💡Okay, so I have this problem about two cyclists starting from two points that are 50 miles apart. When they travel towards each other, they meet in 2 hours. But then, if one cyclist starts 10 miles behind his original position and they travel in opposite directions, they end up 70 miles apart after 3 hours. I need to find the ratio of the speed of the faster cyclist to the slower one. Hmm, let me try to figure this out step by step.First, let's denote the speeds of the two cyclists. Let me call the faster cyclist's speed ( v_1 ) and the slower one's speed ( v_2 ). So, ( v_1 > v_2 ).In the first scenario, they start 50 miles apart and move towards each other, meeting after 2 hours. When two objects move towards each other, their relative speed is the sum of their individual speeds. So, the combined speed is ( v_1 + v_2 ). Since they meet after 2 hours, the total distance covered by both together is 50 miles. So, I can write the equation:[ (v_1 + v_2) times 2 = 50 ]Dividing both sides by 2:[ v_1 + v_2 = 25 ]Okay, so that's our first equation.Now, the second scenario is a bit trickier. One cyclist starts 10 miles behind his original position. So, instead of starting 50 miles apart, they are now 60 miles apart because one is 10 miles behind. But this time, they are moving in opposite directions. So, their relative speed is the difference of their speeds because they are moving away from each other. Wait, no, actually, when moving in opposite directions, their relative speed is the sum of their speeds. Hmm, let me think about that.Wait, no, if they are moving towards each other, their relative speed is the sum, but if they are moving in the same direction, the relative speed is the difference. But in this case, they are moving in opposite directions, so their relative speed should be the sum again because they are moving away from each other. So, the distance between them increases at a rate of ( v_1 + v_2 ).But according to the problem, after 3 hours, they are 70 miles apart. Initially, they were 60 miles apart. So, the distance between them increased by 10 miles over 3 hours. Therefore, the rate at which the distance between them increases is ( frac{10}{3} ) miles per hour. So, that should be equal to their relative speed when moving in opposite directions, which is ( v_1 + v_2 ). Wait, but that can't be because ( v_1 + v_2 ) is 25 mph from the first scenario, and 25 times 3 is 75, which is more than 70. Hmm, maybe I made a mistake here.Wait, let me re-examine. If they are moving in opposite directions, their relative speed is indeed the sum of their speeds. So, starting from 60 miles apart, after 3 hours, the distance between them should be:[ 60 + (v_1 + v_2) times 3 = 70 ]Is that correct? Let me see. If they start 60 miles apart and move away from each other, the distance between them increases by ( (v_1 + v_2) times 3 ). So, the total distance after 3 hours is:[ 60 + 3(v_1 + v_2) = 70 ]Subtracting 60 from both sides:[ 3(v_1 + v_2) = 10 ]Dividing both sides by 3:[ v_1 + v_2 = frac{10}{3} ]Wait, but from the first scenario, we have ( v_1 + v_2 = 25 ). That's a contradiction because 25 is not equal to ( frac{10}{3} ). So, I must have misunderstood the problem.Wait, maybe in the second scenario, they are moving in opposite directions, but one of them is moving in the opposite direction, so their relative speed is actually ( v_1 - v_2 ) if they are moving in opposite directions. Hmm, no, that doesn't make sense. When moving in opposite directions, regardless of their individual directions, their relative speed is the sum of their speeds. So, perhaps I need to reconsider.Wait, maybe the problem is that one cyclist is moving towards the other's starting point, and the other is moving away. So, perhaps their relative speed is ( v_1 - v_2 ). Let me think.Wait, no, if they are moving in opposite directions, regardless of their starting points, their relative speed is the sum. So, perhaps the initial distance is 60 miles, and after 3 hours, the distance is 70 miles. So, the distance increased by 10 miles over 3 hours, so the rate is ( frac{10}{3} ) mph. Therefore, ( v_1 + v_2 = frac{10}{3} ). But that contradicts the first equation where ( v_1 + v_2 = 25 ). So, something is wrong here.Wait, maybe I misinterpreted the second scenario. Let me read it again."If one cyclist starts from a point 10 miles behind his original position and the other remains at his starting position, and they travel in opposite directions, they would be 70 miles apart in 3 hours."So, one cyclist is starting 10 miles behind, so the initial distance between them is 50 + 10 = 60 miles. They are moving in opposite directions, so their relative speed is ( v_1 + v_2 ). After 3 hours, the distance between them is 70 miles. So, the distance increased by 10 miles over 3 hours, so the rate is ( frac{10}{3} ) mph. Therefore, ( v_1 + v_2 = frac{10}{3} ). But that contradicts the first equation where ( v_1 + v_2 = 25 ). So, that can't be right.Wait, maybe the second scenario is not about moving away from each other but moving in opposite directions relative to their starting points. So, perhaps one is moving towards the original starting point, and the other is moving away. So, their relative speed is ( v_1 - v_2 ). Let me try that.So, if one cyclist is moving towards the original starting point, and the other is moving away, their relative speed is ( v_1 - v_2 ). Starting from 60 miles apart, after 3 hours, the distance is 70 miles. So, the distance increased by 10 miles, so:[ 60 + 3(v_1 - v_2) = 70 ]Subtracting 60:[ 3(v_1 - v_2) = 10 ]Dividing by 3:[ v_1 - v_2 = frac{10}{3} ]Okay, that makes sense. So, now we have two equations:1. ( v_1 + v_2 = 25 )2. ( v_1 - v_2 = frac{10}{3} )Now, I can solve these two equations to find ( v_1 ) and ( v_2 ).Adding the two equations:( (v_1 + v_2) + (v_1 - v_2) = 25 + frac{10}{3} )Simplifying:( 2v_1 = frac{75}{3} + frac{10}{3} = frac{85}{3} )So,( v_1 = frac{85}{6} ) mphSubtracting the second equation from the first:( (v_1 + v_2) - (v_1 - v_2) = 25 - frac{10}{3} )Simplifying:( 2v_2 = frac{75}{3} - frac{10}{3} = frac{65}{3} )So,( v_2 = frac{65}{6} ) mphNow, to find the ratio ( frac{v_1}{v_2} ):[ frac{frac{85}{6}}{frac{65}{6}} = frac{85}{65} = frac{17}{13} ]So, the ratio of the faster cyclist's speed to the slower one is ( frac{17}{13} ).Let me double-check my steps to make sure I didn't make any mistakes.1. First scenario: towards each other, meeting in 2 hours. So, combined speed is 25 mph. That seems correct.2. Second scenario: one starts 10 miles behind, so initial distance is 60 miles. They move in opposite directions, so relative speed is ( v_1 - v_2 ) because one is moving towards the original starting point and the other is moving away. After 3 hours, distance is 70 miles, so increase of 10 miles. Thus, ( v_1 - v_2 = frac{10}{3} ). That seems correct.3. Solving the two equations: - ( v_1 + v_2 = 25 ) - ( v_1 - v_2 = frac{10}{3} ) Adding them gives ( 2v_1 = frac{85}{3} ), so ( v_1 = frac{85}{6} ). Subtracting them gives ( 2v_2 = frac{65}{3} ), so ( v_2 = frac{65}{6} ).4. Ratio ( frac{v_1}{v_2} = frac{85}{65} = frac{17}{13} ). That simplifies correctly.Yes, everything checks out. So, the ratio is ( frac{17}{13} ).

💡Okay, so I've got this multiple-choice question about vectors, and I need to figure out which of the given statements are correct. Let me go through each statement one by one and see if I can understand them properly.First, the question lists six statements:① Vectors with the same direction are called equal vectors;② The length of the zero vector is 0;③ Collinear vectors are vectors on the same straight line;④ The zero vector is a vector without direction;⑤ Collinear vectors are not necessarily equal;⑥ Parallel vectors have the same direction.And the options are:A: 2 B: 3 C: 4 D: 5I need to determine which of these statements are correct and then choose the corresponding option.Let me start by recalling some basic concepts about vectors. Vectors have both magnitude (length) and direction. The zero vector is a special case with magnitude zero and no specific direction, as far as I remember. Collinear vectors are vectors that lie on the same line or parallel lines, meaning they can be in the same or opposite directions. Parallel vectors, on the other hand, are vectors that have the same or exactly opposite directions.Now, let's analyze each statement.**Statement ①: Vectors with the same direction are called equal vectors.**Hmm, this seems a bit off. I think for vectors to be equal, they need to have both the same magnitude and the same direction. Just having the same direction isn't enough because their lengths could be different. So, if two vectors are pointing in the same direction but have different magnitudes, they aren't equal. Therefore, I believe statement ① is incorrect.**Statement ②: The length of the zero vector is 0.**This one sounds straightforward. The zero vector is defined as having a magnitude (length) of zero. So, yes, its length is indeed 0. Therefore, statement ② is correct.**Statement ③: Collinear vectors are vectors on the same straight line.**Wait, collinear vectors are vectors that lie on the same line or parallel lines. So, they don't necessarily have to be on the exact same straight line; they can be on parallel lines as well. Therefore, saying they are on the same straight line is too restrictive. So, statement ③ is incorrect.**Statement ④: The zero vector is a vector without direction.**I remember that the zero vector doesn't have a specific direction because its magnitude is zero. So, it's often said to have an undefined or arbitrary direction. Therefore, it's correct to say that the zero vector is a vector without direction. So, statement ④ is correct.**Statement ⑤: Collinear vectors are not necessarily equal.**This makes sense. Collinear vectors can have the same or opposite directions, but their magnitudes can differ. So, even if they are collinear, they aren't necessarily equal vectors. Therefore, statement ⑤ is correct.**Statement ⑥: Parallel vectors have the same direction.**Hmm, parallel vectors are defined as vectors that have the same or exactly opposite directions. So, they don't necessarily have to have the same direction; they can also have opposite directions. Therefore, statement ⑥ is incorrect.So, summarizing my analysis:- ①: Incorrect- ②: Correct- ③: Incorrect- ④: Correct- ⑤: Correct- ⑥: IncorrectTherefore, the correct statements are ②, ④, and ⑤. That means there are 3 correct statements.Looking back at the options:A: 2 B: 3 C: 4 D: 5So, the correct option is B: 3.But wait, let me double-check my analysis to make sure I didn't make any mistakes.For statement ④, I said the zero vector is a vector without direction. That's correct because the zero vector doesn't have a specific direction. It's often considered to have an undefined direction.For statement ⑤, collinear vectors are not necessarily equal. That's correct because collinear vectors can have different magnitudes even if they are in the same or opposite directions.So, yes, ②, ④, and ⑤ are correct, making it 3 correct statements. Therefore, the answer is B: 3.**Final Answer**boxed{B}

💡Okay, so I have this function here: ( f(x) = log left(x + sqrt[3]{1 + x^3}right) ). I need to figure out if it's even, odd, or neither. Hmm, let's recall what even and odd functions are. An even function satisfies ( f(-x) = f(x) ) for all x in its domain, and an odd function satisfies ( f(-x) = -f(x) ). If neither of these conditions hold, then the function is neither even nor odd.Alright, so to determine whether this function is even or odd, I should compute ( f(-x) ) and see how it relates to ( f(x) ). Let's do that step by step.First, let's write down ( f(-x) ):[f(-x) = log left(-x + sqrt[3]{1 + (-x)^3}right)]Simplify the expression inside the cube root:[(-x)^3 = -x^3]So, the expression becomes:[f(-x) = log left(-x + sqrt[3]{1 - x^3}right)]Hmm, that's interesting. Now, I need to see if this expression can be related to ( f(x) ). Let's recall that ( f(x) = log left(x + sqrt[3]{1 + x^3}right) ). So, the only difference between ( f(-x) ) and ( f(x) ) is the sign of the x term and the argument inside the cube root.I wonder if there's a way to manipulate ( f(-x) ) to see if it's equal to ( f(x) ) or ( -f(x) ). Maybe I can rationalize the expression inside the logarithm. Let's try that.Consider the expression inside ( f(-x) ):[-x + sqrt[3]{1 - x^3}]I recall that for expressions of the form ( a + b ), sometimes multiplying by the conjugate can help simplify. However, since we have a cube root, the conjugate would involve terms that can help eliminate the cube root. Let's see.Let me denote ( a = -x ) and ( b = sqrt[3]{1 - x^3} ). Then, the expression is ( a + b ). To rationalize this, I can use the identity:[(a + b)(a^2 - ab + b^2) = a^3 + b^3]But wait, in this case, ( a = -x ) and ( b = sqrt[3]{1 - x^3} ). Let's compute ( a^3 + b^3 ):[(-x)^3 + (sqrt[3]{1 - x^3})^3 = -x^3 + (1 - x^3) = -x^3 + 1 - x^3 = 1 - 2x^3]Hmm, that doesn't seem particularly helpful. Maybe there's another approach.Alternatively, perhaps I can express ( sqrt[3]{1 - x^3} ) in terms of ( sqrt[3]{1 + x^3} ). Let's see:[sqrt[3]{1 - x^3} = sqrt[3]{-(x^3 - 1)} = -sqrt[3]{x^3 - 1}]But that might not directly help either.Wait a minute, let's consider the original function ( f(x) = log left(x + sqrt[3]{1 + x^3}right) ). Maybe if I compute ( f(x) + f(-x) ), I can see if it simplifies to something.Compute ( f(x) + f(-x) ):[log left(x + sqrt[3]{1 + x^3}right) + log left(-x + sqrt[3]{1 - x^3}right)]Using the logarithm property ( log a + log b = log(ab) ), this becomes:[log left( left(x + sqrt[3]{1 + x^3}right) left(-x + sqrt[3]{1 - x^3}right) right)]Now, let's compute the product inside the logarithm:[(x + sqrt[3]{1 + x^3})(-x + sqrt[3]{1 - x^3})]Let me denote ( A = x ), ( B = sqrt[3]{1 + x^3} ), ( C = -x ), and ( D = sqrt[3]{1 - x^3} ). So, the product is ( (A + B)(C + D) ).Expanding this, we get:[AC + AD + BC + BD]Substituting back:[x(-x) + xsqrt[3]{1 - x^3} + sqrt[3]{1 + x^3}(-x) + sqrt[3]{1 + x^3}sqrt[3]{1 - x^3}]Simplify each term:1. ( x(-x) = -x^2 )2. ( xsqrt[3]{1 - x^3} ) remains as is.3. ( sqrt[3]{1 + x^3}(-x) = -xsqrt[3]{1 + x^3} )4. ( sqrt[3]{1 + x^3}sqrt[3]{1 - x^3} = sqrt[3]{(1 + x^3)(1 - x^3)} = sqrt[3]{1 - x^6} )So, putting it all together:[-x^2 + xsqrt[3]{1 - x^3} - xsqrt[3]{1 + x^3} + sqrt[3]{1 - x^6}]This seems complicated. Maybe there's a better way to approach this.Let me try a different strategy. Perhaps I can express ( f(-x) ) in terms of ( f(x) ). Let's recall that ( f(x) = log left(x + sqrt[3]{1 + x^3}right) ). If I can write ( f(-x) ) as ( -f(x) ), then the function is odd.Let's see:[f(-x) = log left(-x + sqrt[3]{1 - x^3}right)]I want to see if this is equal to ( -f(x) ), which would be:[-log left(x + sqrt[3]{1 + x^3}right) = log left( frac{1}{x + sqrt[3]{1 + x^3}} right)]So, if I can show that:[-x + sqrt[3]{1 - x^3} = frac{1}{x + sqrt[3]{1 + x^3}}]then ( f(-x) = -f(x) ), and the function is odd.Let's check if this equality holds. Let me denote ( y = x + sqrt[3]{1 + x^3} ). Then, ( frac{1}{y} = frac{1}{x + sqrt[3]{1 + x^3}} ). I need to see if ( -x + sqrt[3]{1 - x^3} = frac{1}{y} ).Let's compute ( y cdot (-x + sqrt[3]{1 - x^3}) ):[(x + sqrt[3]{1 + x^3})(-x + sqrt[3]{1 - x^3})]This is the same product as before, which we found to be:[-x^2 + xsqrt[3]{1 - x^3} - xsqrt[3]{1 + x^3} + sqrt[3]{1 - x^6}]If ( y cdot (-x + sqrt[3]{1 - x^3}) = 1 ), then ( -x + sqrt[3]{1 - x^3} = frac{1}{y} ). Let's see if this product equals 1.But from the previous expansion, it's clear that the product is not simply 1. It has multiple terms, so unless those terms cancel out to 1, which seems unlikely, this equality doesn't hold. Therefore, my initial assumption that ( f(-x) = -f(x) ) might be incorrect.Wait, maybe I made a mistake in the manipulation. Let me try another approach. Perhaps I can express ( sqrt[3]{1 - x^3} ) in terms of ( sqrt[3]{1 + x^3} ).Let me denote ( u = sqrt[3]{1 + x^3} ). Then, ( u^3 = 1 + x^3 ), so ( x^3 = u^3 - 1 ). Now, let's compute ( sqrt[3]{1 - x^3} ):[sqrt[3]{1 - x^3} = sqrt[3]{1 - (u^3 - 1)} = sqrt[3]{2 - u^3}]Hmm, not sure if that helps.Alternatively, perhaps I can consider the function ( g(x) = x + sqrt[3]{1 + x^3} ). Then, ( f(x) = log(g(x)) ). Let's see if ( g(-x) ) relates to ( g(x) ) in a useful way.Compute ( g(-x) ):[g(-x) = -x + sqrt[3]{1 + (-x)^3} = -x + sqrt[3]{1 - x^3}]Now, let's see if ( g(-x) ) is related to ( 1/g(x) ). If ( g(-x) = 1/g(x) ), then ( f(-x) = log(1/g(x)) = -log(g(x)) = -f(x) ), which would imply that ( f(x) ) is odd.So, let's check if ( g(-x) = 1/g(x) ):[-x + sqrt[3]{1 - x^3} = frac{1}{x + sqrt[3]{1 + x^3}}]Cross-multiplying:[(-x + sqrt[3]{1 - x^3})(x + sqrt[3]{1 + x^3}) = 1]Let's compute the left-hand side:[(-x)(x) + (-x)sqrt[3]{1 + x^3} + sqrt[3]{1 - x^3}(x) + sqrt[3]{1 - x^3}sqrt[3]{1 + x^3}]Simplify each term:1. ( (-x)(x) = -x^2 )2. ( (-x)sqrt[3]{1 + x^3} = -xsqrt[3]{1 + x^3} )3. ( sqrt[3]{1 - x^3}(x) = xsqrt[3]{1 - x^3} )4. ( sqrt[3]{1 - x^3}sqrt[3]{1 + x^3} = sqrt[3]{(1 - x^3)(1 + x^3)} = sqrt[3]{1 - x^6} )So, the left-hand side becomes:[-x^2 - xsqrt[3]{1 + x^3} + xsqrt[3]{1 - x^3} + sqrt[3]{1 - x^6}]This does not simplify to 1 in general, so the equality ( g(-x) = 1/g(x) ) does not hold. Therefore, ( f(-x) neq -f(x) ), which suggests that the function is not odd.Wait, but earlier I thought it might be odd. Maybe I need to reconsider. Let's try plugging in some specific values to test.Let's choose ( x = 1 ):[f(1) = log(1 + sqrt[3]{1 + 1}) = log(1 + sqrt[3]{2}) approx log(1 + 1.26) approx log(2.26) approx 0.354]Now, compute ( f(-1) ):[f(-1) = log(-1 + sqrt[3]{1 - 1}) = log(-1 + sqrt[3]{0}) = log(-1 + 0) = log(-1)]Wait, that's undefined because the logarithm of a negative number is not real. So, ( x = -1 ) is not in the domain of ( f(x) ). Hmm, that's a problem.Let me check the domain of ( f(x) ). The argument of the logarithm must be positive:[x + sqrt[3]{1 + x^3} > 0]Let's analyze this inequality. The cube root function ( sqrt[3]{1 + x^3} ) is defined for all real numbers, but we need the sum ( x + sqrt[3]{1 + x^3} ) to be positive.Let's consider different cases:1. When ( x > 0 ): - ( sqrt[3]{1 + x^3} > sqrt[3]{x^3} = x ), so ( x + sqrt[3]{1 + x^3} > x + x = 2x > 0 ). So, the function is defined for all ( x > 0 ).2. When ( x = 0 ): - ( 0 + sqrt[3]{1 + 0} = 1 > 0 ). So, defined at ( x = 0 ).3. When ( x < 0 ): - Let ( x = -a ) where ( a > 0 ). - Then, ( -a + sqrt[3]{1 + (-a)^3} = -a + sqrt[3]{1 - a^3} ). - We need ( -a + sqrt[3]{1 - a^3} > 0 ). - Let's analyze ( sqrt[3]{1 - a^3} ): - If ( a < 1 ), then ( 1 - a^3 > 0 ), so ( sqrt[3]{1 - a^3} ) is positive. - If ( a = 1 ), ( sqrt[3]{0} = 0 ), so ( -1 + 0 = -1 < 0 ). - If ( a > 1 ), ( 1 - a^3 < 0 ), so ( sqrt[3]{1 - a^3} ) is negative. So, for ( x < 0 ): - If ( |x| < 1 ), ( sqrt[3]{1 - x^3} > 0 ), but we need ( -x + sqrt[3]{1 - x^3} > 0 ). - Let's see if ( sqrt[3]{1 - x^3} > x ). - For small ( x ), say ( x = -0.5 ): [ sqrt[3]{1 - (-0.5)^3} = sqrt[3]{1 + 0.125} = sqrt[3]{1.125} approx 1.04 ] So, ( -(-0.5) + 1.04 = 0.5 + 1.04 = 1.54 > 0 ). So, defined. - For ( x = -1 ): [ sqrt[3]{1 - (-1)^3} = sqrt[3]{1 + 1} = sqrt[3]{2} approx 1.26 ] So, ( -(-1) + 1.26 = 1 + 1.26 = 2.26 > 0 ). Wait, but earlier I thought ( f(-1) ) was undefined. Let me recalculate: [ f(-1) = log(-1 + sqrt[3]{1 - 1}) = log(-1 + 0) = log(-1) ] Oh, that's still undefined. Wait, but according to the domain analysis, ( x = -1 ) should be in the domain because ( -(-1) + sqrt[3]{1 - (-1)^3} = 1 + sqrt[3]{2} > 0 ). So, there must be a mistake in my earlier calculation. Let me recompute ( f(-1) ): [ f(-1) = log(-1 + sqrt[3]{1 + (-1)^3}) = log(-1 + sqrt[3]{0}) = log(-1 + 0) = log(-1) ] Wait, that's still undefined. But according to the domain analysis, ( x = -1 ) should be in the domain because ( -(-1) + sqrt[3]{1 - (-1)^3} = 1 + sqrt[3]{2} > 0 ). So, why is there a discrepancy? Ah, I see the confusion. The function is defined as ( f(x) = log(x + sqrt[3]{1 + x^3}) ). So, for ( x = -1 ): [ f(-1) = log(-1 + sqrt[3]{1 + (-1)^3}) = log(-1 + sqrt[3]{0}) = log(-1 + 0) = log(-1) ] Which is undefined. However, according to the domain analysis, ( x = -1 ) should satisfy ( x + sqrt[3]{1 + x^3} > 0 ). Let's check: [ -1 + sqrt[3]{1 + (-1)^3} = -1 + sqrt[3]{0} = -1 + 0 = -1 < 0 ] So, actually, ( x = -1 ) is not in the domain because the argument of the logarithm is negative. Therefore, my earlier domain analysis was incorrect. Let's correct the domain analysis: For ( x < 0 ): - Let ( x = -a ) where ( a > 0 ). - Then, ( x + sqrt[3]{1 + x^3} = -a + sqrt[3]{1 - a^3} ). - We need ( -a + sqrt[3]{1 - a^3} > 0 ). - Let's analyze ( sqrt[3]{1 - a^3} ): - If ( a < 1 ), ( 1 - a^3 > 0 ), so ( sqrt[3]{1 - a^3} > 0 ). - If ( a = 1 ), ( sqrt[3]{0} = 0 ). - If ( a > 1 ), ( 1 - a^3 < 0 ), so ( sqrt[3]{1 - a^3} < 0 ). So, for ( x < 0 ): - If ( |x| < 1 ), ( sqrt[3]{1 - x^3} > 0 ), and we need ( -x + sqrt[3]{1 - x^3} > 0 ). - For small ( x ), say ( x = -0.5 ): [ -(-0.5) + sqrt[3]{1 - (-0.5)^3} = 0.5 + sqrt[3]{1 + 0.125} = 0.5 + sqrt[3]{1.125} approx 0.5 + 1.04 = 1.54 > 0 ] - So, defined for ( |x| < 1 ). - If ( |x| = 1 ), ( x = -1 ): [ -(-1) + sqrt[3]{1 - (-1)^3} = 1 + sqrt[3]{2} approx 1 + 1.26 = 2.26 > 0 ] Wait, but earlier calculation showed that ( f(-1) = log(-1) ), which is undefined. There's a contradiction here. Wait, no. Let's clarify: - The function is ( f(x) = log(x + sqrt[3]{1 + x^3}) ). - For ( x = -1 ): [ f(-1) = log(-1 + sqrt[3]{1 + (-1)^3}) = log(-1 + sqrt[3]{0}) = log(-1 + 0) = log(-1) ] Which is undefined. - However, the domain analysis suggests that ( x = -1 ) should satisfy ( x + sqrt[3]{1 + x^3} > 0 ), but in reality, it doesn't because ( sqrt[3]{1 + x^3} = sqrt[3]{0} = 0 ), so ( x + sqrt[3]{1 + x^3} = -1 + 0 = -1 < 0 ). Therefore, the domain analysis needs to be corrected. The correct condition is ( x + sqrt[3]{1 + x^3} > 0 ). For ( x = -1 ), this is not satisfied, so ( x = -1 ) is not in the domain. So, for ( x < 0 ), the function is defined only when ( x + sqrt[3]{1 + x^3} > 0 ). Let's find the range of ( x ) for which this holds. Let me define ( h(x) = x + sqrt[3]{1 + x^3} ). We need ( h(x) > 0 ). For ( x > 0 ), as before, ( h(x) > 0 ). For ( x = 0 ), ( h(0) = 0 + sqrt[3]{1} = 1 > 0 ). For ( x < 0 ), let's analyze ( h(x) ): - Let ( x = -a ) where ( a > 0 ). - Then, ( h(x) = -a + sqrt[3]{1 - a^3} ). - We need ( -a + sqrt[3]{1 - a^3} > 0 ). Let's consider ( a ) in different ranges: 1. ( 0 < a < 1 ): - ( 1 - a^3 > 0 ), so ( sqrt[3]{1 - a^3} > 0 ). - Let's see if ( sqrt[3]{1 - a^3} > a ). - For ( a = 0.5 ): [ sqrt[3]{1 - 0.125} = sqrt[3]{0.875} approx 0.957 < 0.5 ] Wait, that's not true. ( 0.957 > 0.5 ), so ( -0.5 + 0.957 approx 0.457 > 0 ). So, defined. - For ( a = 0.8 ): [ sqrt[3]{1 - 0.512} = sqrt[3]{0.488} approx 0.787 < 0.8 ] So, ( -0.8 + 0.787 approx -0.013 < 0 ). Not defined. - Wait, that's contradictory. Let me compute more accurately. For ( a = 0.8 ): [ sqrt[3]{1 - 0.512} = sqrt[3]{0.488} approx 0.787 ] So, ( -0.8 + 0.787 approx -0.013 ), which is less than 0. So, not defined. For ( a = 0.7 ): [ sqrt[3]{1 - 0.343} = sqrt[3]{0.657} approx 0.87 ] So, ( -0.7 + 0.87 approx 0.17 > 0 ). Defined. For ( a = 0.75 ): [ sqrt[3]{1 - 0.421875} = sqrt[3]{0.578125} approx 0.833 ] So, ( -0.75 + 0.833 approx 0.083 > 0 ). Defined. For ( a = 0.78 ): [ sqrt[3]{1 - 0.78^3} approx sqrt[3]{1 - 0.474552} = sqrt[3]{0.525448} approx 0.807 ] So, ( -0.78 + 0.807 approx 0.027 > 0 ). Defined. For ( a = 0.79 ): [ sqrt[3]{1 - 0.79^3} approx sqrt[3]{1 - 0.493039} = sqrt[3]{0.506961} approx 0.797 ] So, ( -0.79 + 0.797 approx 0.007 > 0 ). Defined. For ( a = 0.8 ): [ sqrt[3]{1 - 0.512} = sqrt[3]{0.488} approx 0.787 ] So, ( -0.8 + 0.787 approx -0.013 < 0 ). Not defined. Therefore, there exists some ( a ) between 0.79 and 0.8 where ( h(x) = 0 ). Let's find the exact point where ( h(x) = 0 ): [ -a + sqrt[3]{1 - a^3} = 0 implies sqrt[3]{1 - a^3} = a implies 1 - a^3 = a^3 implies 1 = 2a^3 implies a^3 = frac{1}{2} implies a = sqrt[3]{frac{1}{2}} approx 0.7937 ] So, for ( a < sqrt[3]{frac{1}{2}} approx 0.7937 ), ( h(x) > 0 ), and for ( a > sqrt[3]{frac{1}{2}} ), ( h(x) < 0 ). Therefore, the domain of ( f(x) ) is: - ( x > 0 ) - ( -sqrt[3]{frac{1}{2}} < x < 0 ) - ( x = 0 ) So, ( f(x) ) is defined for ( x > 0 ) and ( x ) in ( (-sqrt[3]{frac{1}{2}}, 0) ). Now, going back to the original question: is ( f(x) ) even, odd, or neither? To determine this, we need to check if ( f(-x) = f(x) ) (even) or ( f(-x) = -f(x) ) (odd) for all ( x ) in the domain. However, the domain of ( f(x) ) is not symmetric about the origin because ( f(x) ) is defined for ( x > 0 ) and ( x ) in ( (-sqrt[3]{frac{1}{2}}, 0) ), but not for ( x < -sqrt[3]{frac{1}{2}} ). Therefore, the function cannot be even or odd because the domain is not symmetric. Wait, but the problem didn't specify the domain, so perhaps we assume the largest possible domain where the function is defined. In that case, the domain is not symmetric, so the function cannot be even or odd. However, the initial problem didn't mention anything about the domain, so maybe we should consider the function as defined for all real numbers where the argument of the logarithm is positive, which, as we saw, is not symmetric. Therefore, the function is neither even nor odd. But wait, earlier I thought it might be odd, but then realized that the domain isn't symmetric. So, perhaps the function is neither even nor odd. Alternatively, maybe there's a way to extend the function to a symmetric domain, but that's not standard practice. Typically, the domain is considered as given by the function's definition. Therefore, considering the domain, the function is neither even nor odd.

💡Alright, let's tackle this problem step by step. I have three points, A(-4, y₁), B(-1, y₂), and C(1, y₃), all lying on the graph of the quadratic function y = x² + 4x + k. I need to figure out the relationship between y₁, y₂, and y₃ and choose the correct option from A to D.First, I remember that quadratic functions graph into parabolas. Since the coefficient of x² is positive (1 in this case), the parabola opens upwards. That means it has a minimum point, called the vertex, and it's symmetric around the vertical line passing through the vertex.To find the vertex, I think I should complete the square for the quadratic function. Let me try that:Starting with y = x² + 4x + k.To complete the square, I take the coefficient of x, which is 4, divide it by 2 to get 2, and then square it to get 4. So, I can rewrite the equation as:y = (x² + 4x + 4) + (k - 4)Which simplifies to:y = (x + 2)² + (k - 4)So, the vertex form of the quadratic is y = (x + 2)² + (k - 4). This tells me the vertex is at (-2, k - 4). The axis of symmetry is the vertical line x = -2.Now, since the parabola opens upwards, the further a point is from the vertex along the axis of symmetry, the larger its y-value will be. Points on either side of the vertex will have increasing y-values as they move away from the vertex.Looking at the points given:- Point A is at x = -4, which is 2 units to the left of the vertex at x = -2.- Point B is at x = -1, which is 1 unit to the right of the vertex.- Point C is at x = 1, which is 3 units to the right of the vertex.Wait, hold on. I think I made a mistake there. Let me recast that.Actually, the distance from the vertex is calculated as the absolute difference from x = -2.So:- For point A at x = -4: | -4 - (-2) | = | -2 | = 2 units.- For point B at x = -1: | -1 - (-2) | = | 1 | = 1 unit.- For point C at x = 1: | 1 - (-2) | = | 3 | = 3 units.So, point A is 2 units to the left, point B is 1 unit to the right, and point C is 3 units to the right of the vertex.Since the parabola opens upwards, the further a point is from the vertex, the higher its y-value. So, the order from smallest to largest y-value should be based on their distances from the vertex.Point B is closest to the vertex (1 unit away), so it should have the smallest y-value. Then, point A is 2 units away, so it should have a larger y-value than B. Finally, point C is 3 units away, so it should have the largest y-value.Therefore, the order is y₂ < y₁ < y₃.Let me double-check by plugging in the x-values into the original equation to find the y-values.For point A(-4, y₁):y₁ = (-4)² + 4*(-4) + k = 16 - 16 + k = 0 + k = kFor point B(-1, y₂):y₂ = (-1)² + 4*(-1) + k = 1 - 4 + k = -3 + kFor point C(1, y₃):y₃ = (1)² + 4*(1) + k = 1 + 4 + k = 5 + kSo, y₁ = k, y₂ = k - 3, y₃ = k + 5.Comparing these:y₂ = k - 3 < y₁ = k < y₃ = k + 5So, y₂ < y₁ < y₃, which matches what I concluded earlier.Therefore, the correct option is B: y₂ < y₁ < y₃.

💡First, I need to determine the amount of land that is clay-rich and the amount that is the other type of soil. Since one-third of Jorge's 60 acres is clay-rich, I calculate:[frac{1}{3} times 60 = 20 text{ acres}]This means the remaining two-thirds is the other type of soil:[frac{2}{3} times 60 = 40 text{ acres}]Next, I let ( Y ) represent the yield of the other type of soil in bushels per acre. Given that clay-rich soil yields half as much, the yield for clay-rich soil is ( frac{Y}{2} ) bushels per acre.I then calculate the total yield from each type of soil:- From clay-rich soil (20 acres):[20 times frac{Y}{2} = 10Y text{ bushels}]- From the other type of soil (40 acres):[40 times Y = 40Y text{ bushels}]Adding these together gives the total yield:[10Y + 40Y = 20,000 text{ bushels}]Combining like terms:[50Y = 20,000]Solving for ( Y ):[Y = frac{20,000}{50} = 400 text{ bushels per acre}]

💡Okay, so I need to figure out which three students received an A in physics class based on their statements. There are four students: Emily, Frank, Grace, and Harry. Each of them made a conditional statement about getting an A. Also, we know that exactly three of them received an A. Let me try to break this down step by step.First, let's list out what each student said:1. **Emily** said, "If I get an A, then Frank will also get an A."2. **Frank** said, "If I get an A, then Grace will also get an A."3. **Grace** said, "If I get an A, then Harry will also get an A."4. **Harry** said, "If I get an A, then Emily cannot get an A."So, these are all conditional statements. In logic, these are implications. For example, Emily's statement can be written as E → F, where E is "Emily gets an A" and F is "Frank gets an A." Similarly, Frank's statement is F → G, Grace's is G → H, and Harry's is H → ¬E.Given that exactly three students received an A, we need to figure out which three it could be without violating any of their statements.Let me consider each possibility one by one.**Option A: Emily, Frank, Grace**If Emily got an A, then according to her statement, Frank must also get an A. That's consistent. Frank getting an A means Grace gets an A, which is also consistent. Grace getting an A means Harry should get an A, but in this option, Harry doesn't get an A. So, Grace's statement is violated because if she got an A, Harry should also get an A. Therefore, this option is invalid.**Option B: Frank, Grace, Harry**Let's see. If Frank got an A, then Grace must get an A, which is fine. Grace getting an A means Harry must get an A, which is also fine. Now, Harry got an A, so according to his statement, Emily cannot get an A. That's consistent because Emily isn't in this group. So, this option seems to hold without any contradictions.**Option C: Grace, Harry, Emily**Grace got an A, so Harry must get an A, which is fine. Harry got an A, so Emily cannot get an A. But in this option, Emily does get an A, which contradicts Harry's statement. Therefore, this option is invalid.**Option D: Grace, Emily, Frank**Grace got an A, so Harry must get an A, but Harry isn't in this group. That's a contradiction because Grace's statement requires Harry to get an A if she does. So, this option is invalid.**Option E: Harry, Emily, Frank**Harry got an A, so Emily cannot get an A. But in this option, Emily does get an A, which contradicts Harry's statement. Therefore, this option is invalid.So, after evaluating all the options, only **Option B: Frank, Grace, Harry** doesn't lead to any contradictions and satisfies all the given statements.