💡Okay, so I have this inequality to prove: for each non-negative integer ( n ) and real number ( x ), the expression ( sin{x} cdot (n sin{x} - sin{nx}) ) is greater than or equal to zero. Hmm, that sounds a bit tricky, but let me try to break it down.First, I know that ( sin{x} ) is a periodic function with period ( 2pi ), and it oscillates between -1 and 1. The term ( n sin{x} ) is just scaling ( sin{x} ) by the integer ( n ), so it will also oscillate between -n and n. The term ( sin{nx} ) is another sine function, but with a frequency that's ( n ) times that of ( sin{x} ). So, as ( n ) increases, ( sin{nx} ) oscillates more rapidly.I need to show that the product of ( sin{x} ) and ( (n sin{x} - sin{nx}) ) is non-negative. That means either both ( sin{x} ) and ( (n sin{x} - sin{nx}) ) are non-negative, or both are non-positive.Maybe I can consider different cases based on the value of ( x ). Since ( sin{x} ) can be positive or negative, perhaps I should analyze the expression when ( sin{x} ) is positive and when it's negative separately.Let me start by assuming ( sin{x} geq 0 ). In this case, for the entire product to be non-negative, ( (n sin{x} - sin{nx}) ) must also be non-negative. So, I need to show that ( n sin{x} geq sin{nx} ) when ( sin{x} geq 0 ).Similarly, if ( sin{x} leq 0 ), then ( (n sin{x} - sin{nx}) ) must also be less than or equal to zero. So, ( n sin{x} leq sin{nx} ) when ( sin{x} leq 0 ).Hmm, okay. Maybe I can use some trigonometric identities to simplify ( sin{nx} ). I remember that ( sin{nx} ) can be expressed using multiple-angle formulas. For example, ( sin{2x} = 2 sin{x} cos{x} ), ( sin{3x} = 3 sin{x} - 4 sin^3{x} ), and so on. Maybe there's a general expression for ( sin{nx} ) in terms of ( sin{x} ).Wait, I think there's a recursive formula for ( sin{nx} ). Let me recall. I believe it's something like ( sin{nx} = 2 cos{x} sin{(n-1)x} - sin{(n-2)x} ). Yeah, that sounds right. So, perhaps I can use induction on ( n ) to prove the inequality.Let me try mathematical induction. For ( n = 0 ), the expression becomes ( sin{x} cdot (0 - sin{0}) = sin{x} cdot 0 = 0 ), which satisfies the inequality. For ( n = 1 ), it's ( sin{x} cdot (sin{x} - sin{x}) = sin{x} cdot 0 = 0 ), which is also non-negative.Now, assume that the inequality holds for some integer ( k ), meaning ( sin{x} cdot (k sin{x} - sin{kx}) geq 0 ). I need to show it holds for ( k + 1 ). So, let's consider ( sin{x} cdot ((k + 1) sin{x} - sin{(k + 1)x}) ).Using the recursive formula for ( sin{(k + 1)x} ), which is ( 2 cos{x} sin{kx} - sin{(k - 1)x} ). Substituting this in, we get:( sin{x} cdot ((k + 1) sin{x} - (2 cos{x} sin{kx} - sin{(k - 1)x})) )Simplifying inside the parentheses:( (k + 1) sin{x} - 2 cos{x} sin{kx} + sin{(k - 1)x} )Hmm, this seems a bit complicated. Maybe I should express ( sin{(k + 1)x} ) in terms of ( sin{kx} ) and ( sin{(k - 1)x} ) and see if I can relate it back to the inductive hypothesis.Alternatively, maybe I can use another approach. Let's consider the function ( f(n) = n sin{x} - sin{nx} ). I need to show that ( sin{x} cdot f(n) geq 0 ).Taking the derivative of ( f(n) ) with respect to ( n ), we get ( f'(n) = sin{x} - x cos{nx} ). Hmm, not sure if that helps directly, but perhaps analyzing the behavior of ( f(n) ) as ( n ) increases could give some insight.Wait, another idea: maybe using the identity for ( sin{nx} ) in terms of ( sin{x} ) and multiple angles. For example, ( sin{nx} ) can be written as a sum involving ( sin{x} ) and powers of ( cos{x} ). But that might get messy.Let me think about specific values of ( x ) to see if I can spot a pattern or get some intuition. For example, when ( x = 0 ), ( sin{0} = 0 ), so the entire expression is zero, which is fine. When ( x = pi/2 ), ( sin{pi/2} = 1 ), and ( sin{n cdot pi/2} ) alternates between 1, 0, -1, etc., depending on ( n ). So, ( n sin{x} = n ), and ( sin{nx} ) is either 1, 0, or -1. So, ( n sin{x} - sin{nx} ) is ( n - 1 ), ( n ), or ( n + 1 ), all of which are positive since ( n ) is non-negative. Thus, ( sin{x} cdot (n sin{x} - sin{nx}) ) is positive.Similarly, when ( x = pi ), ( sin{pi} = 0 ), so the expression is zero. When ( x = 3pi/2 ), ( sin{3pi/2} = -1 ), and ( sin{n cdot 3pi/2} ) alternates similarly. So, ( n sin{x} = -n ), and ( sin{nx} ) is either -1, 0, or 1. Thus, ( n sin{x} - sin{nx} ) is ( -n - (-1) = -n + 1 ), ( -n ), or ( -n - 1 ). Since ( n ) is non-negative, these are all less than or equal to zero. Therefore, ( sin{x} cdot (n sin{x} - sin{nx}) ) is non-negative because both factors are negative or zero.Hmm, so in these specific cases, the inequality holds. Maybe I can generalize this behavior.Another approach: consider the function ( f(n) = n sin{x} - sin{nx} ). Let's analyze its sign. If ( sin{x} geq 0 ), then ( n sin{x} ) is non-negative. Now, ( sin{nx} ) can be positive or negative, but perhaps ( n sin{x} ) is always greater than or equal to ( sin{nx} ) when ( sin{x} geq 0 ).Wait, is that true? Let's see. For ( x ) in ( [0, pi] ), ( sin{x} ) is non-negative. Let's pick ( x = pi/4 ). Then ( sin{x} = sqrt{2}/2 approx 0.707 ). For ( n = 1 ), ( sin{nx} = sin{pi/4} = sqrt{2}/2 ), so ( n sin{x} - sin{nx} = 0 ). For ( n = 2 ), ( sin{2x} = sin{pi/2} = 1 ), so ( 2 cdot sqrt{2}/2 - 1 = sqrt{2} - 1 approx 0.414 ), which is positive. For ( n = 3 ), ( sin{3x} = sin{3pi/4} = sqrt{2}/2 ), so ( 3 cdot sqrt{2}/2 - sqrt{2}/2 = 2sqrt{2}/2 = sqrt{2} approx 1.414 ), positive. For ( n = 4 ), ( sin{4x} = sin{pi} = 0 ), so ( 4 cdot sqrt{2}/2 - 0 = 2sqrt{2} approx 2.828 ), positive.So, in this case, ( n sin{x} - sin{nx} ) is non-negative for ( n geq 1 ). Similarly, for ( x = pi/3 ), ( sin{x} = sqrt{3}/2 approx 0.866 ). For ( n = 1 ), ( sin{nx} = sqrt{3}/2 ), so ( n sin{x} - sin{nx} = 0 ). For ( n = 2 ), ( sin{2x} = sin{2pi/3} = sqrt{3}/2 ), so ( 2 cdot sqrt{3}/2 - sqrt{3}/2 = sqrt{3}/2 approx 0.866 ), positive. For ( n = 3 ), ( sin{3x} = sin{pi} = 0 ), so ( 3 cdot sqrt{3}/2 - 0 = 3sqrt{3}/2 approx 2.598 ), positive.This seems to suggest that when ( sin{x} geq 0 ), ( n sin{x} - sin{nx} geq 0 ). Similarly, when ( sin{x} leq 0 ), ( n sin{x} - sin{nx} leq 0 ), making the product non-negative.But how can I prove this in general? Maybe using induction as I thought earlier.Let's try induction again. Base case: ( n = 0 ). The expression is ( sin{x} cdot (0 - sin{0}) = 0 ), which is non-negative. For ( n = 1 ), it's ( sin{x} cdot (sin{x} - sin{x}) = 0 ), also non-negative.Assume it's true for ( n = k ), so ( sin{x} cdot (k sin{x} - sin{kx}) geq 0 ). Now, consider ( n = k + 1 ). We need to show ( sin{x} cdot ((k + 1) sin{x} - sin{(k + 1)x}) geq 0 ).Using the identity ( sin{(k + 1)x} = sin{kx + x} = sin{kx}cos{x} + cos{kx}sin{x} ). Substituting this in:( sin{x} cdot ((k + 1)sin{x} - (sin{kx}cos{x} + cos{kx}sin{x})) )Simplify inside the parentheses:( (k + 1)sin{x} - sin{kx}cos{x} - cos{kx}sin{x} )Factor out ( sin{x} ):( sin{x}(k + 1 - cos{kx}) - sin{kx}cos{x} )Hmm, not sure if that helps. Maybe rearrange terms:( k sin{x} + sin{x} - sin{kx}cos{x} - cos{kx}sin{x} )Group terms:( k sin{x} - sin{kx}cos{x} + sin{x} - cos{kx}sin{x} )Factor ( sin{x} ) from the last two terms:( k sin{x} - sin{kx}cos{x} + sin{x}(1 - cos{kx}) )Now, from the inductive hypothesis, ( sin{x} cdot (k sin{x} - sin{kx}) geq 0 ). Let's denote ( A = sin{x} cdot (k sin{x} - sin{kx}) geq 0 ).Looking back at our expression:( k sin{x} - sin{kx}cos{x} + sin{x}(1 - cos{kx}) )Notice that ( k sin{x} - sin{kx}cos{x} = sin{x} cdot k - sin{kx}cos{x} ). Maybe factor ( sin{x} ) out of the first term:( sin{x} cdot k - sin{kx}cos{x} = sin{x} cdot (k - frac{sin{kx}cos{x}}{sin{x}}) )But that might not be helpful. Alternatively, perhaps express ( sin{kx} ) in terms of ( sin{(k - 1)x} ) and ( sin{(k + 1)x} ) using the recursive identity.Wait, I think I need a different approach. Maybe consider the function ( f(n) = n sin{x} - sin{nx} ) and analyze its behavior as ( n ) increases.For ( sin{x} geq 0 ), as ( n ) increases, ( n sin{x} ) increases linearly, while ( sin{nx} ) oscillates between -1 and 1. So, for sufficiently large ( n ), ( n sin{x} ) will dominate, making ( f(n) ) positive. But I need to show it's non-negative for all ( n geq 0 ).Wait, perhaps using the fact that ( sin{nx} leq n sin{x} ) when ( sin{x} geq 0 ). Is that a known inequality? I'm not sure, but maybe I can prove it.Consider ( x ) in ( [0, pi] ) where ( sin{x} geq 0 ). For ( n = 1 ), ( sin{x} = sin{x} ), so equality holds. For ( n = 2 ), ( sin{2x} = 2 sin{x} cos{x} ). Since ( cos{x} leq 1 ), ( sin{2x} leq 2 sin{x} ). So, ( 2 sin{x} - sin{2x} geq 0 ).Similarly, for ( n = 3 ), ( sin{3x} = 3 sin{x} - 4 sin^3{x} ). So, ( 3 sin{x} - sin{3x} = 4 sin^3{x} geq 0 ) since ( sin{x} geq 0 ).For ( n = 4 ), ( sin{4x} = 2 sin{2x} cos{2x} = 4 sin{x} cos{x} cos{2x} ). So, ( 4 sin{x} - sin{4x} = 4 sin{x} (1 - cos{x} cos{2x}) ). Since ( cos{x} cos{2x} leq 1 ), this is non-negative.Hmm, this seems to hold for these specific ( n ). Maybe there's a pattern here. Let me try to generalize.Assume that for some ( k geq 1 ), ( k sin{x} geq sin{kx} ) when ( sin{x} geq 0 ). Then, for ( k + 1 ), using the identity ( sin{(k + 1)x} = sin{kx}cos{x} + cos{kx}sin{x} ), we have:( (k + 1)sin{x} - sin{(k + 1)x} = (k + 1)sin{x} - (sin{kx}cos{x} + cos{kx}sin{x}) )Simplify:( k sin{x} + sin{x} - sin{kx}cos{x} - cos{kx}sin{x} )Group terms:( k sin{x} - sin{kx}cos{x} + sin{x}(1 - cos{kx}) )From the inductive hypothesis, ( k sin{x} geq sin{kx} ). Since ( cos{x} leq 1 ), ( sin{kx}cos{x} leq sin{kx} ). Therefore, ( k sin{x} - sin{kx}cos{x} geq k sin{x} - sin{kx} geq 0 ) by the inductive hypothesis.Also, ( 1 - cos{kx} geq 0 ) because ( cos{kx} leq 1 ). Since ( sin{x} geq 0 ), the term ( sin{x}(1 - cos{kx}) geq 0 ).Therefore, both terms in the expression are non-negative, so their sum is non-negative. Thus, ( (k + 1)sin{x} - sin{(k + 1)x} geq 0 ).By induction, this holds for all ( n geq 0 ) when ( sin{x} geq 0 ).Similarly, when ( sin{x} leq 0 ), we can use a similar argument. Since ( sin{x} leq 0 ), ( n sin{x} leq 0 ). Also, ( sin{nx} ) will oscillate, but we need to show ( n sin{x} - sin{nx} leq 0 ).Using the same approach, for ( sin{x} leq 0 ), ( n sin{x} leq 0 ), and ( sin{nx} ) can be positive or negative. However, using the identity and similar steps as above, we can show that ( n sin{x} - sin{nx} leq 0 ) when ( sin{x} leq 0 ).Thus, in both cases, the product ( sin{x} cdot (n sin{x} - sin{nx}) ) is non-negative, which completes the proof.

💡Okay, so I need to find the value of ( f(10sqrt{3}) ) given that ( f(x) ) is an odd function with some specific properties. Let me try to break this down step by step.First, I know that ( f(x) ) is an odd function. That means ( f(-x) = -f(x) ) for all real numbers ( x ). This property might come in handy later, especially when dealing with negative arguments or when I need to relate positive and negative values of the function.Next, the problem states that for any real number ( x ), ( f(x + 2) = -f(x) ). Hmm, this seems like a functional equation that relates the value of the function at ( x + 2 ) to its value at ( x ). I wonder if this implies some kind of periodicity? Let me check:If I apply the given equation twice, I get:[ f(x + 4) = f((x + 2) + 2) = -f(x + 2) = -(-f(x)) = f(x) ]So, ( f(x + 4) = f(x) ). That means the function ( f(x) ) is periodic with period 4. That's useful because it allows me to reduce the argument of the function modulo 4 when dealing with large values like ( 10sqrt{3} ).Alright, so ( f(x) ) is periodic with period 4. Let me note that down:[ f(x + 4) = f(x) ]Also, the function is defined on the interval ( [0, 1] ) as ( f(x) = 2x ). Since it's an odd function, I can extend this definition to the interval ( [-1, 0] ) by using the property ( f(-x) = -f(x) ). So, for ( x in [-1, 0] ), ( f(x) = -2x ).Now, to find ( f(10sqrt{3}) ), I can use the periodicity of the function. Since the period is 4, I can subtract multiples of 4 from ( 10sqrt{3} ) until I get a value within the interval ( [0, 4) ). Let me calculate ( 10sqrt{3} ) first to get an approximate idea.Calculating ( 10sqrt{3} ):[ sqrt{3} approx 1.732 ][ 10sqrt{3} approx 10 times 1.732 = 17.32 ]So, ( 10sqrt{3} approx 17.32 ). Now, I need to find ( 17.32 mod 4 ) to reduce it within the interval ( [0, 4) ).Calculating ( 17.32 div 4 ):[ 4 times 4 = 16 ][ 17.32 - 16 = 1.32 ]So, ( 17.32 mod 4 approx 1.32 ). Therefore, ( f(10sqrt{3}) = f(1.32) ).Now, I need to find ( f(1.32) ). Since ( 1.32 ) is between 1 and 2, I need to figure out how to express ( f(x) ) in the interval ( [1, 2] ). The function is defined on ( [0, 1] ) as ( 2x ), but for ( x ) beyond 1, I need to use the given functional equation ( f(x + 2) = -f(x) ).Let me express ( 1.32 ) in terms of a value within ( [0, 1] ). Let me set ( x = 1.32 - 2 = -0.68 ). Wait, that's negative. But since the function is odd, I can relate ( f(-0.68) ) to ( f(0.68) ).So, ( f(1.32) = f(-0.68 + 2) = -f(-0.68) ) by the functional equation ( f(x + 2) = -f(x) ). But since ( f ) is odd, ( f(-0.68) = -f(0.68) ). Therefore:[ f(1.32) = -f(-0.68) = -(-f(0.68)) = f(0.68) ]Wait, hold on, that seems contradictory. Let me double-check.Given ( f(x + 2) = -f(x) ), so ( f(1.32) = f(-0.68 + 2) = -f(-0.68) ). Since ( f ) is odd, ( f(-0.68) = -f(0.68) ). Therefore:[ f(1.32) = -f(-0.68) = -(-f(0.68)) = f(0.68) ]But ( 0.68 ) is within ( [0, 1] ), so we can directly compute ( f(0.68) ) using the given definition:[ f(0.68) = 2 times 0.68 = 1.36 ]Therefore, ( f(1.32) = 1.36 ). But wait, earlier I thought ( f(1.32) = f(0.68) ), but actually, according to the functional equation, ( f(1.32) = -f(-0.68) = f(0.68) ). So, ( f(1.32) = 1.36 ).But hold on, let me think again. If ( f(x + 2) = -f(x) ), then ( f(1.32) = -f(1.32 - 2) = -f(-0.68) ). Since ( f ) is odd, ( f(-0.68) = -f(0.68) ). Therefore, ( f(1.32) = -(-f(0.68)) = f(0.68) ). So, yes, ( f(1.32) = f(0.68) = 1.36 ).Wait, but I'm getting ( f(1.32) = 1.36 ), but the original function is defined as ( f(x) = 2x ) on ( [0, 1] ), and we've extended it using the functional equation. But is this correct?Let me try another approach. Since ( f(x + 2) = -f(x) ), then ( f(x + 4) = f(x) ). So, the function is periodic with period 4. So, ( f(10sqrt{3}) = f(10sqrt{3} - 4 times 4) = f(10sqrt{3} - 16) ).Calculating ( 10sqrt{3} - 16 ):[ 10sqrt{3} approx 17.32 ][ 17.32 - 16 = 1.32 ]So, ( f(10sqrt{3}) = f(1.32) ).Now, ( 1.32 ) is in the interval ( [1, 2] ). To find ( f(1.32) ), we can use the functional equation ( f(x + 2) = -f(x) ). Let me set ( x = 1.32 - 2 = -0.68 ). So, ( f(1.32) = f(-0.68 + 2) = -f(-0.68) ).Since ( f ) is odd, ( f(-0.68) = -f(0.68) ). Therefore, ( f(1.32) = -(-f(0.68)) = f(0.68) ).Now, ( 0.68 ) is in ( [0, 1] ), so ( f(0.68) = 2 times 0.68 = 1.36 ).Therefore, ( f(1.32) = 1.36 ), and so ( f(10sqrt{3}) = 1.36 ).Wait, but earlier I thought ( f(1.32) = -f(-0.68) = f(0.68) ), which is positive. But let me check if I made a mistake in the sign.Given ( f(x + 2) = -f(x) ), so ( f(1.32) = f(-0.68 + 2) = -f(-0.68) ). Since ( f(-0.68) = -f(0.68) ), then ( f(1.32) = -(-f(0.68)) = f(0.68) ). So, yes, it's positive.But wait, let me think about the interval ( [1, 2] ). If ( x ) is in ( [1, 2] ), then ( x - 2 ) is in ( [-1, 0] ). So, ( f(x) = -f(x - 2) ). Since ( x - 2 ) is negative, ( f(x - 2) = -f(2 - x) ) because ( f ) is odd. Therefore, ( f(x) = -(-f(2 - x)) = f(2 - x) ).So, for ( x in [1, 2] ), ( f(x) = f(2 - x) ). Since ( 2 - x ) is in ( [0, 1] ), we can use the given definition ( f(2 - x) = 2(2 - x) ).Wait, that's different from what I did earlier. Let me clarify.If ( x in [1, 2] ), then ( x - 2 in [-1, 0] ). So, ( f(x) = -f(x - 2) ). But ( x - 2 ) is negative, so ( f(x - 2) = -f(2 - x) ) because ( f ) is odd. Therefore, ( f(x) = -(-f(2 - x)) = f(2 - x) ).Since ( 2 - x ) is in ( [0, 1] ), ( f(2 - x) = 2(2 - x) ). Therefore, ( f(x) = 2(2 - x) ) for ( x in [1, 2] ).So, for ( x = 1.32 ), ( f(1.32) = 2(2 - 1.32) = 2(0.68) = 1.36 ). So, that's consistent with what I found earlier.But wait, this seems conflicting with the initial approach where I thought ( f(1.32) = f(0.68) ). But actually, both approaches lead to the same result because ( f(2 - x) = 2(2 - x) ), which is the same as ( f(0.68) = 2 times 0.68 ).Wait, no. Let me clarify:If ( x = 1.32 ), then ( 2 - x = 0.68 ). So, ( f(1.32) = f(0.68) ). But ( f(0.68) = 2 times 0.68 = 1.36 ). So, yes, both methods give the same result.Therefore, ( f(1.32) = 1.36 ), so ( f(10sqrt{3}) = 1.36 ).But wait, let me think again. The function is defined as ( f(x) = 2x ) on ( [0, 1] ), and for ( x in [1, 2] ), ( f(x) = 2(2 - x) ). So, that's a V-shaped function peaking at ( x = 1 ) with ( f(1) = 2(2 - 1) = 2 ).Wait, but if ( x = 1 ), then ( f(1) = 2(2 - 1) = 2 ). But according to the original definition, ( f(1) = 2 times 1 = 2 ). So, it's consistent at ( x = 1 ).Similarly, at ( x = 0.5 ), ( f(0.5) = 2 times 0.5 = 1 ), and at ( x = 1.5 ), ( f(1.5) = 2(2 - 1.5) = 2 times 0.5 = 1 ). So, the function is symmetric around ( x = 1 ) in the interval ( [0, 2] ).But wait, since the function is odd, it should be symmetric about the origin. However, the definition on ( [0, 1] ) is linear, and on ( [1, 2] ) it's also linear but decreasing. So, the function is piecewise linear on ( [0, 2] ), and then it repeats every 4 units.Wait, but the function is defined on all real numbers, so beyond ( [0, 2] ), it's defined using the functional equation ( f(x + 2) = -f(x) ). So, for ( x in [2, 3] ), ( f(x) = -f(x - 2) ). Since ( x - 2 in [0, 1] ), ( f(x) = -2(x - 2) ).Similarly, for ( x in [3, 4] ), ( f(x) = -f(x - 2) ). Since ( x - 2 in [1, 2] ), ( f(x - 2) = 2(2 - (x - 2)) = 2(4 - x) ). Therefore, ( f(x) = -2(4 - x) = 2(x - 4) ).So, putting it all together, the function ( f(x) ) has the following definition on ( [0, 4] ):- ( [0, 1] ): ( f(x) = 2x )- ( [1, 2] ): ( f(x) = 2(2 - x) )- ( [2, 3] ): ( f(x) = -2(x - 2) )- ( [3, 4] ): ( f(x) = 2(x - 4) )And since the function is periodic with period 4, this pattern repeats every 4 units.Now, going back to ( f(10sqrt{3}) approx f(1.32) ). Since ( 1.32 ) is in ( [1, 2] ), we use the definition ( f(x) = 2(2 - x) ).Calculating ( f(1.32) ):[ f(1.32) = 2(2 - 1.32) = 2(0.68) = 1.36 ]Therefore, ( f(10sqrt{3}) = 1.36 ).But wait, earlier I thought ( f(1.32) = f(0.68) ), which is also 1.36. So, both methods confirm that ( f(1.32) = 1.36 ).However, I need to make sure about the sign. Since ( f(x + 2) = -f(x) ), and ( f(x) ) is odd, does that affect the sign in any way?Wait, let me re-examine the steps:1. ( f(10sqrt{3}) = f(10sqrt{3} - 4 times 4) = f(10sqrt{3} - 16) approx f(1.32) ).2. ( 1.32 in [1, 2] ), so ( f(1.32) = 2(2 - 1.32) = 1.36 ).Alternatively, using the functional equation:1. ( f(1.32) = f(-0.68 + 2) = -f(-0.68) ).2. Since ( f ) is odd, ( f(-0.68) = -f(0.68) ).3. Therefore, ( f(1.32) = -(-f(0.68)) = f(0.68) ).4. ( f(0.68) = 2 times 0.68 = 1.36 ).Both methods give the same result, so I think the answer is correct.But just to be thorough, let me plot the function mentally. From ( x = 0 ) to ( x = 1 ), it's a straight line from 0 to 2. From ( x = 1 ) to ( x = 2 ), it's a straight line decreasing back to 0. From ( x = 2 ) to ( x = 3 ), it's a straight line going negative, from 0 to -2. From ( x = 3 ) to ( x = 4 ), it's a straight line increasing back to 0.So, at ( x = 1.32 ), which is between 1 and 2, the function is decreasing from 2 to 0, so the value should be positive and less than 2. Since 1.32 is 0.32 units above 1, the function decreases by 0.64 (since the slope is -2), so ( f(1.32) = 2 - 0.64 = 1.36 ). That matches our earlier calculation.Therefore, I'm confident that ( f(10sqrt{3}) = 1.36 ).But wait, let me check if I made a mistake in the periodicity. The function has period 4, so ( f(x + 4) = f(x) ). So, ( f(10sqrt{3}) = f(10sqrt{3} - 4 times 4) = f(10sqrt{3} - 16) approx f(1.32) ). That's correct.Alternatively, I could have considered that ( 10sqrt{3} ) is approximately 17.32, and 17.32 divided by 4 is 4.33, so the remainder is 1.32. So, yes, ( f(10sqrt{3}) = f(1.32) ).Therefore, the final answer is ( boxed{1.36} ).Wait, but in the initial problem statement, it's mentioned that ( f(x) ) is an odd function. So, if ( f(10sqrt{3}) = 1.36 ), is that consistent with the function being odd?Wait, ( 10sqrt{3} ) is a positive number, so ( f(10sqrt{3}) ) should be positive if the function is positive in that interval. Since ( 1.32 ) is in ( [1, 2] ), and the function is positive there, it's consistent.But let me think about the function's behavior beyond ( [0, 4] ). For example, at ( x = 5 ), which is ( 1 ) more than ( 4 ), ( f(5) = f(1) = 2 ). Similarly, ( f(6) = f(2) = 0 ), ( f(7) = f(3) = -2 ), ( f(8) = f(4) = 0 ), and so on. So, the function repeats every 4 units, which makes sense.Therefore, I think my answer is correct. ( f(10sqrt{3}) = 1.36 ).But wait, let me double-check the calculation of ( 10sqrt{3} mod 4 ). ( 10sqrt{3} approx 17.32 ). Dividing 17.32 by 4 gives 4.33, so the integer part is 4, and the remainder is ( 17.32 - 4 times 4 = 17.32 - 16 = 1.32 ). So, yes, ( 10sqrt{3} mod 4 approx 1.32 ).Therefore, ( f(10sqrt{3}) = f(1.32) = 1.36 ).I think I've considered all possible angles and checked my work multiple times. I don't see any mistakes in the reasoning. So, I'm confident that the answer is ( boxed{1.36} ).

💡Okay, so I have this problem here: there's a right cylinder with a base radius of 4 units inscribed in a sphere with a radius of 6 units. I need to find the volume inside the sphere but outside the cylinder, and express it as ( Zpi ), then find ( Z ) as a common fraction. Hmm, okay, let me think about how to approach this.First, I know that the cylinder is inscribed in the sphere, which means the cylinder touches the sphere exactly at the top and bottom, and around the curved surface. So, the sphere's diameter should be related to the cylinder's height and radius somehow.The sphere has a radius of 6 units, so its diameter is 12 units. That diameter should be equal to the space diagonal of the cylinder, right? Because if you imagine the cylinder inside the sphere, the longest distance inside the sphere would pass through the center of the cylinder and touch the top and bottom of the cylinder.Let me draw a mental picture. The cylinder has a radius of 4 units, so the base is a circle with radius 4. The height of the cylinder, let's call it ( h ), is something I need to find. Since the cylinder is inscribed, the sphere's diameter is the diagonal of the cylinder. So, if I consider a right triangle formed by the radius of the cylinder's base, half the height of the cylinder, and the sphere's radius, I can use the Pythagorean theorem.Wait, let me clarify. The sphere's radius is 6, so the distance from the center of the sphere to any point on its surface is 6. The cylinder is inscribed, so the center of the sphere is also the center of the cylinder. Therefore, the distance from the center of the sphere to the top or bottom of the cylinder is half the height of the cylinder, which is ( h/2 ). The radius of the cylinder's base is 4 units, so in this right triangle, one leg is 4, another leg is ( h/2 ), and the hypotenuse is 6.So, applying the Pythagorean theorem: ( 4^2 + (h/2)^2 = 6^2 ). Let me write that down:( 16 + (h^2)/4 = 36 )Subtracting 16 from both sides:( (h^2)/4 = 20 )Multiplying both sides by 4:( h^2 = 80 )Taking the square root:( h = sqrt{80} )Simplify ( sqrt{80} ):( sqrt{80} = sqrt{16 times 5} = 4sqrt{5} )So, the height of the cylinder is ( 4sqrt{5} ) units.Now, I need to find the volume inside the sphere but outside the cylinder. That means I need to subtract the volume of the cylinder from the volume of the sphere.First, let's compute the volume of the sphere. The formula for the volume of a sphere is ( frac{4}{3}pi r^3 ). Here, the radius ( r ) is 6 units.So,( V_{sphere} = frac{4}{3}pi (6)^3 )Calculating ( 6^3 ):( 6 times 6 = 36 )( 36 times 6 = 216 )So,( V_{sphere} = frac{4}{3}pi times 216 )Multiplying ( frac{4}{3} times 216 ):First, divide 216 by 3: ( 216 / 3 = 72 )Then multiply by 4: ( 72 times 4 = 288 )So,( V_{sphere} = 288pi )Alright, now the volume of the cylinder. The formula for the volume of a cylinder is ( pi r^2 h ). Here, the radius ( r ) is 4 units, and the height ( h ) is ( 4sqrt{5} ) units.So,( V_{cylinder} = pi (4)^2 (4sqrt{5}) )Calculating ( 4^2 ):( 4 times 4 = 16 )Then,( 16 times 4sqrt{5} = 64sqrt{5} )So,( V_{cylinder} = 64sqrt{5}pi )Now, the volume inside the sphere but outside the cylinder is the difference between the two volumes:( V_{space} = V_{sphere} - V_{cylinder} )Plugging in the values:( V_{space} = 288pi - 64sqrt{5}pi )Factor out ( pi ):( V_{space} = (288 - 64sqrt{5})pi )So, the total volume is ( (288 - 64sqrt{5})pi ), which is equal to ( Zpi ). Therefore, ( Z = 288 - 64sqrt{5} ).Wait, but the problem says to express ( Z ) as a common fraction. Hmm, 288 and 64 are both integers, but ( sqrt{5} ) is irrational. So, is ( Z ) supposed to be a fraction, or is it just expressed as a combination of integers and radicals? The problem says "as a common fraction," so maybe I need to represent it differently.Wait, perhaps I made a mistake in interpreting the problem. Let me check again.The problem says: "The total volume, in cubic units, of the space inside the sphere and outside the cylinder is ( Zpi ). Find ( Z ), as a common fraction."Hmm, so ( Z ) is just the coefficient of ( pi ), which is ( 288 - 64sqrt{5} ). But 288 and 64 are integers, so ( Z ) is ( 288 - 64sqrt{5} ). But the problem says to express ( Z ) as a common fraction. Maybe I need to factor something out?Looking at 288 and 64, both are divisible by 16. Let's see:288 divided by 16 is 18, and 64 divided by 16 is 4. So,( 288 - 64sqrt{5} = 16 times (18 - 4sqrt{5}) )But that doesn't really help in terms of making it a common fraction. Alternatively, maybe I need to rationalize or something? Wait, no, because ( Z ) is just a coefficient, not necessarily a fraction itself.Wait, perhaps I misread the problem. Let me go back."A right cylinder with a base radius of 4 units is inscribed in a sphere of radius 6 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is ( Zpi ). Find ( Z ), as a common fraction."So, ( Z ) is the coefficient, which is ( 288 - 64sqrt{5} ). But the problem says to express ( Z ) as a common fraction. Hmm, maybe I need to write it as a single fraction? But ( 288 - 64sqrt{5} ) is already simplified, and it's not a fraction. Wait, unless I factor out something else.Alternatively, maybe I made a mistake in calculating the volume of the cylinder. Let me double-check.Cylinder volume: ( pi r^2 h ). ( r = 4 ), ( h = 4sqrt{5} ). So,( pi times 16 times 4sqrt{5} = 64sqrt{5}pi ). That seems correct.Sphere volume: ( frac{4}{3}pi r^3 ), ( r = 6 ). So,( frac{4}{3}pi times 216 = 288pi ). That also seems correct.So, the difference is indeed ( 288pi - 64sqrt{5}pi = (288 - 64sqrt{5})pi ). Therefore, ( Z = 288 - 64sqrt{5} ).But the problem says to express ( Z ) as a common fraction. Hmm, perhaps I need to write it as a fraction with denominator 1? But that doesn't make sense. Alternatively, maybe I need to rationalize or represent it differently.Wait, perhaps I need to express ( Z ) as a fraction in terms of the sphere's volume or something. Let me think.Alternatively, maybe I need to compute ( Z ) as a fraction of the sphere's volume? But the problem doesn't specify that. It just says the total volume is ( Zpi ), so ( Z ) is just the coefficient.Wait, maybe the problem expects ( Z ) to be a fraction without radicals? But ( 288 - 64sqrt{5} ) is not a rational number, so it can't be expressed as a common fraction in the traditional sense. Maybe I made a mistake earlier in calculating the height of the cylinder.Let me go back to that step.We had the sphere radius 6, cylinder radius 4. So, the relationship is:( (cylinder radius)^2 + (half cylinder height)^2 = (sphere radius)^2 )So,( 4^2 + (h/2)^2 = 6^2 )Which is,( 16 + (h^2)/4 = 36 )Subtract 16,( (h^2)/4 = 20 )Multiply by 4,( h^2 = 80 )So,( h = sqrt{80} = 4sqrt{5} )That seems correct.Alternatively, maybe I need to express the height as a fraction? But ( 4sqrt{5} ) is already simplified.Wait, perhaps the problem expects the answer in terms of fractions without radicals, but that doesn't seem possible because the height involves a radical. So, maybe the answer is just ( 288 - 64sqrt{5} ), and since it's already in terms of integers and radicals, that's the value of ( Z ).But the problem says "as a common fraction," which usually implies a fraction with integers in numerator and denominator, like ( frac{a}{b} ). So, maybe I need to write ( Z ) as a fraction where the numerator and denominator are integers, but that would require expressing ( sqrt{5} ) as a fraction, which isn't possible because it's irrational.Wait, perhaps I made a mistake in interpreting the problem. Maybe the cylinder is inscribed such that its height is equal to the sphere's diameter? But no, because if the cylinder's height were 12, then the radius would have to be zero, which contradicts the given radius of 4.Alternatively, maybe I need to consider the cylinder's height differently. Wait, no, the approach with the Pythagorean theorem seems correct.Alternatively, maybe the problem is expecting the answer in terms of a fraction multiplied by ( pi ), but ( Z ) itself is just a number, not necessarily a fraction. Hmm.Wait, looking back at the problem statement: "The total volume, in cubic units, of the space inside the sphere and outside the cylinder is ( Zpi ). Find ( Z ), as a common fraction."So, ( Z ) is just the coefficient of ( pi ), which is ( 288 - 64sqrt{5} ). Since ( Z ) is supposed to be a common fraction, but ( 288 - 64sqrt{5} ) is not a fraction, perhaps I need to express it differently.Wait, maybe I need to factor out a common factor from 288 and 64. Let's see:288 and 64 are both divisible by 16.288 ÷ 16 = 1864 ÷ 16 = 4So,( 288 - 64sqrt{5} = 16(18 - 4sqrt{5}) )But that still doesn't make it a common fraction. Alternatively, maybe I need to write it as a fraction over 1:( Z = frac{288 - 64sqrt{5}}{1} )But that's trivial and doesn't really help.Alternatively, perhaps I made a mistake in calculating the volumes. Let me double-check.Sphere volume: ( frac{4}{3}pi r^3 ). ( r = 6 ), so ( 6^3 = 216 ). ( frac{4}{3} times 216 = 288 ). So, ( 288pi ). Correct.Cylinder volume: ( pi r^2 h ). ( r = 4 ), so ( 4^2 = 16 ). ( h = 4sqrt{5} ). So, ( 16 times 4sqrt{5} = 64sqrt{5} ). So, ( 64sqrt{5}pi ). Correct.Difference: ( 288pi - 64sqrt{5}pi = (288 - 64sqrt{5})pi ). So, ( Z = 288 - 64sqrt{5} ).Hmm, I think that's the correct value for ( Z ). Since the problem asks for ( Z ) as a common fraction, but ( Z ) is an expression involving an integer and a radical, perhaps the answer is simply ( 288 - 64sqrt{5} ), and that's acceptable as a common fraction in the context of the problem, even though it's not a simple fraction with numerator and denominator.Alternatively, maybe I need to rationalize or present it differently, but I don't see a way to do that without complicating it further.Wait, perhaps I can write ( Z ) as a fraction by expressing it over 1:( Z = frac{288 - 64sqrt{5}}{1} )But that's just the same as ( 288 - 64sqrt{5} ), so it doesn't really help.Alternatively, maybe the problem expects ( Z ) to be expressed in terms of fractions without radicals, but that's not possible because the height involves a radical.Alternatively, perhaps I made a mistake in the initial step of using the Pythagorean theorem. Let me think again.The sphere has radius 6, so diameter 12. The cylinder is inscribed, so the diagonal of the cylinder (from top to bottom through the center) is equal to the sphere's diameter, which is 12. So, the diagonal of the cylinder is 12.The cylinder has radius 4, so the distance from the center to the side is 4. The height of the cylinder is ( h ), so half the height is ( h/2 ). So, the diagonal of the cylinder is the hypotenuse of a right triangle with legs 4 and ( h/2 ), and hypotenuse 6 (since the sphere's radius is 6, not diameter). Wait, hold on, I think I made a mistake here.Wait, the sphere's radius is 6, so the distance from the center to any point on the sphere is 6. The cylinder is inscribed, so the top and bottom of the cylinder touch the sphere. Therefore, the distance from the center of the sphere to the top of the cylinder is ( h/2 ), and the radius of the cylinder's base is 4. So, the right triangle has legs 4 and ( h/2 ), and hypotenuse 6 (the sphere's radius). So, the equation is:( 4^2 + (h/2)^2 = 6^2 )Which is:( 16 + (h^2)/4 = 36 )Subtract 16:( (h^2)/4 = 20 )Multiply by 4:( h^2 = 80 )So,( h = sqrt{80} = 4sqrt{5} )So, that part is correct.Wait, but earlier I thought the hypotenuse was 6, which is correct because the sphere's radius is 6. So, the equation is correct.Therefore, the height is indeed ( 4sqrt{5} ), and the volumes are correct.So, perhaps the answer is just ( Z = 288 - 64sqrt{5} ), and that's acceptable as a common fraction in the context of the problem, even though it's not a simple fraction. Alternatively, maybe the problem expects the answer in terms of a fraction with numerator and denominator, but since ( Z ) is a combination of integers and radicals, it can't be expressed as a simple fraction.Alternatively, perhaps I need to write ( Z ) as a fraction by factoring out something else. Let me see:( 288 - 64sqrt{5} = 16(18 - 4sqrt{5}) )But that's still not a common fraction. Alternatively, maybe I need to express it as a mixed number or something, but that doesn't make sense here.Wait, perhaps the problem is expecting the answer in terms of the sphere's volume, but no, the problem just says ( Zpi ).Alternatively, maybe I need to rationalize the expression, but ( 288 - 64sqrt{5} ) is already simplified.Wait, perhaps I made a mistake in calculating the volumes. Let me check again.Sphere volume: ( frac{4}{3}pi (6)^3 = frac{4}{3}pi times 216 = 288pi ). Correct.Cylinder volume: ( pi (4)^2 (4sqrt{5}) = pi times 16 times 4sqrt{5} = 64sqrt{5}pi ). Correct.Difference: ( 288pi - 64sqrt{5}pi = (288 - 64sqrt{5})pi ). So, ( Z = 288 - 64sqrt{5} ).Hmm, I think that's the correct answer. Since the problem asks for ( Z ) as a common fraction, but ( Z ) is an expression involving an integer and a radical, perhaps the answer is simply ( 288 - 64sqrt{5} ), and that's acceptable as a common fraction in the context of the problem, even though it's not a simple fraction with numerator and denominator.Alternatively, maybe the problem expects ( Z ) to be expressed in terms of fractions without radicals, but that's not possible because the height involves a radical.Wait, perhaps I need to write ( Z ) as a fraction by expressing it over 1:( Z = frac{288 - 64sqrt{5}}{1} )But that's just the same as ( 288 - 64sqrt{5} ), so it doesn't really help.Alternatively, maybe the problem expects ( Z ) to be expressed in terms of fractions without radicals, but that's not possible because the height involves a radical.I think I've double-checked all my steps, and the calculations seem correct. So, perhaps the answer is simply ( Z = 288 - 64sqrt{5} ), and that's the value they're looking for, even though it's not a simple fraction. Maybe the term "common fraction" was used loosely here, or perhaps it's a misinterpretation.Alternatively, maybe I need to express ( Z ) as a fraction in terms of the sphere's volume or something else, but the problem doesn't specify that.Wait, another thought: perhaps I need to rationalize the expression ( 288 - 64sqrt{5} ) in some way, but I don't see how that would help in making it a common fraction.Alternatively, maybe I need to write ( Z ) as a fraction where both terms are fractions, but that doesn't make sense because ( 288 ) and ( 64sqrt{5} ) are separate terms.Wait, perhaps I can write ( Z ) as ( frac{288}{1} - frac{64sqrt{5}}{1} ), but that's just the same as ( 288 - 64sqrt{5} ).Alternatively, maybe I need to factor out a common factor from both terms. Let's see:288 and 64 are both divisible by 16, as I noted earlier. So,( 288 - 64sqrt{5} = 16(18 - 4sqrt{5}) )But that still doesn't make it a common fraction. Alternatively, maybe I can write it as ( frac{16(18 - 4sqrt{5})}{1} ), but again, that's not helpful.Alternatively, maybe I need to express ( Z ) as a fraction where the numerator is ( 288 - 64sqrt{5} ) and the denominator is 1, but that's trivial.Wait, perhaps the problem expects ( Z ) to be expressed as a fraction in terms of the sphere's volume, but that's not indicated.Alternatively, maybe I need to consider that ( Z ) is a fraction of the sphere's volume, but the problem states that the volume outside the cylinder is ( Zpi ), so ( Z ) is just the coefficient.I think I've exhausted all possibilities. The calculations seem correct, and the answer is ( Z = 288 - 64sqrt{5} ). Since the problem asks for ( Z ) as a common fraction, but ( Z ) is an expression involving an integer and a radical, perhaps the answer is simply ( 288 - 64sqrt{5} ), and that's acceptable.Alternatively, maybe I made a mistake in the initial step of using the Pythagorean theorem. Let me think again.The sphere has radius 6, so the distance from the center to the top of the cylinder is ( h/2 ), and the radius of the cylinder's base is 4. So, the right triangle has legs 4 and ( h/2 ), and hypotenuse 6. So, the equation is:( 4^2 + (h/2)^2 = 6^2 )Which is:( 16 + (h^2)/4 = 36 )Subtract 16:( (h^2)/4 = 20 )Multiply by 4:( h^2 = 80 )So,( h = sqrt{80} = 4sqrt{5} )Yes, that's correct.So, I think I've confirmed all steps, and the answer is ( Z = 288 - 64sqrt{5} ). Therefore, the final answer is ( boxed{288 - 64sqrt{5}} ).

💡First, I need to determine the actual lengths of the sides of the rectangular field based on the given ratio of 3:4. I'll represent the sides as 3x and 4x, where x is a common multiplier.Next, I'll use the area of the field to find the value of x. The area of a rectangle is calculated by multiplying its length and width, so I'll set up the equation:10092 = (3x) × (4x)Simplifying this equation will allow me to solve for x.After finding the value of x, I'll calculate the actual lengths of the sides by multiplying 3x and 4x by the value of x.Once I have the lengths of both sides, I'll determine the perimeter of the rectangle, which is the total length of fencing needed. The perimeter of a rectangle is calculated as 2 times the sum of its length and width.Finally, I'll calculate the cost per meter of fencing by dividing the total cost of fencing by the total perimeter of the field.

💡Okay, so I have this problem about folding a rectangular piece of paper and then punching a hole in it. I need to figure out what the paper will look like when it's unfolded. Let me try to break it down step by step.First, the paper is folded left to right. I'm picturing taking a standard rectangular sheet, like a piece of printer paper, and folding it so that the left edge meets the right edge. This should halve the width of the paper. So after this first fold, the paper is now half as wide as it was originally.Next, the paper is folded diagonally from the top-left corner to the bottom-right corner. Okay, so after the first fold, the paper is half its original width, and now I'm folding it diagonally. This means the top-left corner will meet the bottom-right corner, effectively creating a diagonal crease from one corner to the opposite corner. This fold should divide the paper into two triangular sections.Now, a hole is punched near the center of this folded paper. Since the paper has been folded twice, punching a hole near the center will create multiple holes when the paper is unfolded. I need to figure out how many holes there will be and where they'll be located.Let me visualize this. After folding left to right, the paper is half its original width. Then folding it diagonally creates two triangles. Punching a hole near the center would mean punching through all layers at that point. When I unfold the diagonal fold first, the hole will be mirrored along the diagonal line, resulting in two holes. Then, when I unfold the left-to-right fold, each of those two holes will be mirrored again, resulting in four holes in total.So, when the paper is fully unfolded, there should be four holes. Now, where exactly will these holes be located? Since the first fold was left to right, the holes will be symmetric about the vertical center line. The second fold was diagonal, so the holes will also be symmetric about the diagonal line.Let me double-check this. If I fold left to right, any hole punched will have a mirror image on the other side of the vertical center line. Then, folding diagonally will create a mirror image along the diagonal. So, punching one hole will result in four holes when both folds are unfolded.I think that makes sense. So, the paper will have four holes symmetrically placed about both the vertical center line and the diagonal line.I don't think it's just two holes because the diagonal fold would also create a mirror image. And it's definitely not one hole because we're unfolding both folds, which would replicate the hole multiple times.So, the answer should be B) Four holes symmetrically placed about the diagonal and vertical center lines.

💡Okay, so I have this function f(x) which is defined as f(x) = x squared minus 2x plus 1. And the question is asking me to find the value of f(4). Alright, let me think about how to approach this.First, I remember that when you're given a function like f(x) and you need to find f(4), it means you substitute x with 4 in the function and then calculate the result. So, I need to replace every x in the equation with the number 4.Let me write that down step by step. The function is f(x) = x² - 2x + 1. So, substituting 4 for x, it becomes f(4) = (4)² - 2*(4) + 1. Now, I need to compute each part of this equation.Starting with the first term, (4) squared. I know that squaring a number means multiplying it by itself. So, 4 times 4 is 16. Got that down.Next, the second term is -2 times 4. Hmm, multiplying 2 by 4 gives me 8, and since there's a negative sign in front, it becomes -8. Okay, so that's straightforward.Now, the last term is just +1. So, putting it all together, I have 16 minus 8 plus 1. Let me compute that step by step to avoid any mistakes.First, 16 minus 8. That's 8. Then, adding 1 to that result. So, 8 plus 1 equals 9. Therefore, f(4) should be 9.Wait, let me double-check my calculations to make sure I didn't make any errors. So, 4 squared is definitely 16. Then, 2 times 4 is 8, and with the negative sign, it's -8. Adding 1 at the end. So, 16 minus 8 is 8, and 8 plus 1 is 9. Yep, that seems correct.Alternatively, I can think about the function f(x) = x² - 2x + 1 in a different way. Maybe I can factor it to see if it simplifies, which might help me understand why f(4) is 9.Looking at the quadratic equation x² - 2x + 1, I notice that it's a perfect square trinomial. Because (x - 1) squared equals x² - 2x + 1. So, f(x) can be rewritten as (x - 1)². That might make substitution easier.So, if f(x) = (x - 1)², then f(4) would be (4 - 1) squared. That's 3 squared, which is 9. So, that confirms my earlier result. Both methods give me the same answer, which is reassuring.I guess another way to look at it is by plugging in x = 4 into the original function without factoring. Let me do that again to ensure consistency.So, f(4) = (4)² - 2*(4) + 1. Calculating each term:- 4 squared is 16.- 2 times 4 is 8, and with the negative sign, it's -8.- The constant term is +1.Adding them up: 16 - 8 + 1. 16 minus 8 is 8, and 8 plus 1 is 9. Yep, same result.I wonder if there's another perspective or method to solve this, just to be thorough. Maybe using calculus or something, but that seems unnecessary for such a simple function. Calculus would involve derivatives or integrals, which aren't required here since we're just evaluating the function at a specific point.Alternatively, I could create a table of values for f(x) around x = 4 to see how the function behaves, but that might be overcomplicating things. For example:- f(3) = 9 - 6 + 1 = 4- f(4) = 16 - 8 + 1 = 9- f(5) = 25 - 10 + 1 = 16Looking at these values, it seems like the function is increasing as x increases beyond its vertex. The vertex of this parabola is at x = 1, which is the minimum point since the coefficient of x² is positive. So, as we move to the right of x = 1, the function values increase, which makes sense why f(4) is larger than f(3).But again, that's more about understanding the function's behavior rather than just finding f(4). Since the question only asks for f(4), I don't need to go into that level of detail.Another thought: maybe I can use the concept of function transformation. The original function f(x) = x² is a standard parabola. The function f(x) = x² - 2x + 1 can be seen as a transformation of this standard parabola.Specifically, f(x) = x² - 2x + 1 can be rewritten as f(x) = (x - 1)², as I did earlier. This represents a horizontal shift of the standard parabola f(x) = x² to the right by 1 unit. So, the vertex, which was at (0,0), is now at (1,0). Therefore, when x is 4, which is 3 units to the right of the vertex, the function value is 3² = 9. That's consistent with my previous calculations.I think I've covered multiple methods to arrive at the same answer, which is 9. Each method reinforces the correctness of the solution. I don't see any mistakes in my reasoning, and all approaches lead to the same result. So, I'm confident that f(4) is indeed 9.