💡Okay, so I have this problem here: I need to figure out how many matches can be made from a wooden cube that's 1 meter on each edge. Each match has to be 5 centimeters long, 2 millimeters wide, and 2 millimeters high. And I'm told to assume that no material is lost in the cutting process. Alright, let's break this down step by step.First off, I need to understand the dimensions of the cube and the matches. The cube is 1 meter on each side. I know that 1 meter is equal to 100 centimeters, so the cube is 100 cm x 100 cm x 100 cm. That seems straightforward.Now, each match is 5 cm long, 2 mm wide, and 2 mm high. Hmm, the units here are mixed—centimeters and millimeters. I think it would be best to convert everything to the same unit to make calculations easier. Since the cube is measured in centimeters, I'll convert the millimeters to centimeters. There are 10 millimeters in a centimeter, so 2 mm is 0.2 cm. That means each match is 5 cm long, 0.2 cm wide, and 0.2 cm high.Next, I need to find out the volume of the cube and the volume of one match. Volume is calculated by multiplying length x width x height. For the cube, that would be 100 cm x 100 cm x 100 cm. Let me do that: 100 x 100 is 10,000, and 10,000 x 100 is 1,000,000. So the cube has a volume of 1,000,000 cubic centimeters.Now, for one match, the volume would be 5 cm x 0.2 cm x 0.2 cm. Let me calculate that: 5 x 0.2 is 1, and 1 x 0.2 is 0.2. So each match has a volume of 0.2 cubic centimeters.To find out how many matches can be made from the cube, I need to divide the total volume of the cube by the volume of one match. That would be 1,000,000 cm³ divided by 0.2 cm³ per match. Let me do that division: 1,000,000 divided by 0.2. Hmm, dividing by 0.2 is the same as multiplying by 5, right? Because 1 divided by 0.2 is 5. So, 1,000,000 x 5 is 5,000,000.Wait a second, that seems like a lot of matches. Is that right? Let me double-check my calculations. The cube is 100 cm on each side, so 100 x 100 x 100 is indeed 1,000,000 cm³. Each match is 5 cm long, 0.2 cm wide, and 0.2 cm high, so 5 x 0.2 x 0.2 is 0.2 cm³. Dividing 1,000,000 by 0.2 gives 5,000,000. Yeah, that seems correct.But just to be thorough, maybe I should think about the dimensions in another way. If I have a cube that's 100 cm on each side, how many matches can I line up along each edge? For the length, each match is 5 cm, so along the 100 cm edge, I can fit 100 / 5 = 20 matches. For the width and height, each match is 0.2 cm, so along the 100 cm edge, I can fit 100 / 0.2 = 500 matches.So, in terms of arranging the matches within the cube, I can fit 20 along the length, 500 along the width, and 500 along the height. To find the total number of matches, I multiply these together: 20 x 500 x 500. Let's calculate that: 20 x 500 is 10,000, and 10,000 x 500 is 5,000,000. Okay, that matches my earlier calculation.So, both methods give me the same result: 5,000,000 matches. That seems like a huge number, but when you think about it, each match is very small compared to the cube, so it makes sense that you can fit millions of them inside.I guess another way to think about it is in terms of scaling. If I have a cube that's 100 cm on each side and matches that are 5 cm long, 0.2 cm wide, and 0.2 cm high, the cube is 20 times longer than a match, 500 times wider, and 500 times taller. So, the total number of matches is 20 x 500 x 500, which is indeed 5,000,000.I don't see any mistakes in my reasoning. The key was to convert all measurements to the same unit and then either calculate the volumes or think about how many matches fit along each dimension. Both approaches led me to the same answer, so I'm confident that 5,000,000 is the correct number of matches.

💡First, I need to determine the total amount of automobile installment credit, which is 20% of the total consumer installment credit of 855 billion.Next, I'll calculate 20% of 855 billion to find the automobile installment credit.After obtaining the automobile installment credit, I'll find out how much of this was extended by automobile finance companies, which is one-third of the automobile installment credit.Finally, I'll compute one-third of the automobile installment credit to get the amount of credit extended by automobile finance companies.

💡Okay, so I have this polynomial: ( x^3 - 7x^2 + 14x - 8 ). I need to find all its roots. Hmm, I remember that for polynomials, roots are the values of ( x ) that make the polynomial equal to zero. So, I need to solve the equation ( x^3 - 7x^2 + 14x - 8 = 0 ).First, I think I should try to factor this polynomial because if I can factor it, I can easily find the roots by setting each factor equal to zero. Factoring a cubic polynomial can be a bit tricky, but I remember something called the Rational Root Theorem. Let me recall what that is. The Rational Root Theorem says that any possible rational root of the polynomial ( a_nx^n + dots + a_1x + a_0 ) is of the form ( frac{p}{q} ), where ( p ) is a factor of the constant term ( a_0 ) and ( q ) is a factor of the leading coefficient ( a_n ).In my polynomial, the constant term is ( -8 ) and the leading coefficient is ( 1 ) because the ( x^3 ) term has a coefficient of 1. So, the possible rational roots are the factors of ( -8 ) divided by the factors of ( 1 ). Since the factors of ( 1 ) are just ( pm1 ), the possible rational roots are ( pm1, pm2, pm4, pm8 ).Alright, so I have these possible roots: 1, -1, 2, -2, 4, -4, 8, -8. I need to test each of these to see if they make the polynomial equal to zero.Let me start with ( x = 1 ):( 1^3 - 7(1)^2 + 14(1) - 8 = 1 - 7 + 14 - 8 = 0 ). Oh, that works! So, ( x = 1 ) is a root.Now, let's try ( x = 2 ):( 2^3 - 7(2)^2 + 14(2) - 8 = 8 - 28 + 28 - 8 = 0 ). That also works! So, ( x = 2 ) is another root.Let me try ( x = 4 ) just in case:( 4^3 - 7(4)^2 + 14(4) - 8 = 64 - 112 + 56 - 8 = 0 ). Wait, that's also zero! So, ( x = 4 ) is another root. Hmm, but wait, the polynomial is cubic, so it should have three roots, right? But I already found three roots: 1, 2, and 4. But when I tried ( x = 2 ), it worked, and then ( x = 4 ) also worked. That seems a bit confusing because I thought maybe there could be multiplicities, but let me check my calculations again.Wait, when I plugged in ( x = 4 ), I got:( 64 - 112 + 56 - 8 = 0 ). Let me compute that step by step:64 - 112 is -48, then -48 + 56 is 8, and 8 - 8 is 0. So, yes, ( x = 4 ) is also a root. But that would mean the polynomial has three distinct roots: 1, 2, and 4. But earlier, when I factored it, I thought it might have a repeated root at 2. Maybe I made a mistake in factoring.Let me try to factor the polynomial using the roots I found. Since ( x = 1 ) is a root, I can factor out ( (x - 1) ). Let's perform polynomial division or use synthetic division to factor it out.Using synthetic division with root 1:The coefficients are 1 (for ( x^3 )), -7 (for ( x^2 )), 14 (for ( x )), and -8 (constant term).Bring down the 1.Multiply 1 by 1 (the root) and add to -7: 1*1 = 1; -7 + 1 = -6.Multiply -6 by 1: -6*1 = -6; add to 14: 14 + (-6) = 8.Multiply 8 by 1: 8*1 = 8; add to -8: -8 + 8 = 0. Perfect, no remainder.So, after factoring out ( (x - 1) ), the polynomial becomes ( (x - 1)(x^2 - 6x + 8) ).Now, let's factor ( x^2 - 6x + 8 ). Looking for two numbers that multiply to 8 and add up to -6. Those numbers are -2 and -4 because (-2)*(-4)=8 and (-2)+(-4)=-6. So, ( x^2 - 6x + 8 = (x - 2)(x - 4) ).Therefore, the complete factorization is ( (x - 1)(x - 2)(x - 4) ). So, the roots are 1, 2, and 4, each with multiplicity 1. Wait, but earlier when I thought I had a repeated root at 2, that must have been a mistake.Let me double-check by expanding ( (x - 1)(x - 2)(x - 4) ) to see if it matches the original polynomial.First, multiply ( (x - 1)(x - 2) ):( (x - 1)(x - 2) = x^2 - 2x - x + 2 = x^2 - 3x + 2 ).Now, multiply that by ( (x - 4) ):( (x^2 - 3x + 2)(x - 4) = x^3 - 4x^2 - 3x^2 + 12x + 2x - 8 = x^3 - 7x^2 + 14x - 8 ).Yes, that matches the original polynomial. So, my initial thought about a repeated root at 2 was incorrect. The polynomial actually has three distinct roots: 1, 2, and 4.Wait, but earlier when I tested ( x = 2 ), I got zero, and then when I tested ( x = 4 ), I also got zero. So, both are roots. Therefore, the polynomial has three distinct roots: 1, 2, and 4.But in my first attempt, I thought ( x = 2 ) was a repeated root, but that was a mistake because when I factored correctly, I got three distinct roots.So, to summarize, the roots of the polynomial ( x^3 - 7x^2 + 14x - 8 ) are 1, 2, and 4, each occurring once.Wait, but let me check once more. Maybe I made a mistake in the synthetic division or factoring.Let me try synthetic division again with root 2:Coefficients: 1, -7, 14, -8Bring down the 1.Multiply 1 by 2: 2; add to -7: -5.Multiply -5 by 2: -10; add to 14: 4.Multiply 4 by 2: 8; add to -8: 0.So, after factoring out ( (x - 2) ), the polynomial becomes ( (x - 2)(x^2 - 5x + 4) ).Now, factor ( x^2 - 5x + 4 ): looking for two numbers that multiply to 4 and add to -5. Those are -1 and -4.So, ( x^2 - 5x + 4 = (x - 1)(x - 4) ).Therefore, the complete factorization is ( (x - 2)(x - 1)(x - 4) ), which is the same as before.So, the roots are indeed 1, 2, and 4, each with multiplicity 1.Wait, but earlier I thought I had a repeated root at 2, but that was a mistake. So, the correct roots are 1, 2, and 4.But let me check the original polynomial at x=2 again:( 2^3 - 7(2)^2 + 14(2) - 8 = 8 - 28 + 28 - 8 = 0 ). Yes, that's correct.And at x=4:( 4^3 - 7(4)^2 + 14(4) - 8 = 64 - 112 + 56 - 8 = 0 ). Correct.So, all three are roots, and each occurs once.Therefore, the roots are 1, 2, and 4.But wait, in the initial problem, the user mentioned that if a root occurs more than once, I should enter it as many times as its multiplicity. So, in this case, since all roots are distinct, I just list them once each.So, the final answer should be 1, 2, 4.But wait, in the initial assistant's answer, they thought there was a repeated root at 2, but that was incorrect. So, I need to make sure I don't make that mistake.Wait, let me check the factorization again. If I factor ( x^3 - 7x^2 + 14x - 8 ), I get ( (x - 1)(x - 2)(x - 4) ), which are all distinct factors. So, no repeated roots.Therefore, the roots are 1, 2, and 4, each with multiplicity 1.So, the final answer is 1, 2, 4.But wait, in the initial problem, the user wrote the polynomial as ( x^3 - 7x^2 + 14x - 8 ). Let me double-check that I didn't misread it.Yes, it's ( x^3 - 7x^2 + 14x - 8 ).So, my conclusion is that the roots are 1, 2, and 4.But wait, let me check one more time by plugging in x=1, x=2, and x=4 into the polynomial.For x=1: 1 - 7 + 14 - 8 = 0. Correct.For x=2: 8 - 28 + 28 - 8 = 0. Correct.For x=4: 64 - 112 + 56 - 8 = 0. Correct.So, all three are indeed roots, and each is a simple root (multiplicity 1).Therefore, the roots are 1, 2, and 4.But wait, in the initial assistant's answer, they thought there was a repeated root at 2, but that was a mistake. So, I need to make sure I don't make that mistake.Wait, let me see. If I had a repeated root at 2, the factorization would be ( (x - 1)(x - 2)^2 ). But when I factored it correctly, I got ( (x - 1)(x - 2)(x - 4) ), which means no repeated roots.So, the correct roots are 1, 2, and 4, each occurring once.Therefore, the final answer is 1, 2, 4.But wait, let me check the original polynomial again to make sure I didn't misread it.Yes, it's ( x^3 - 7x^2 + 14x - 8 ).So, I think I'm confident that the roots are 1, 2, and 4.But just to be thorough, let me try to graph the polynomial or at least analyze its behavior to see if it crosses the x-axis three times, which would confirm three distinct real roots.The leading term is ( x^3 ), so as ( x ) approaches positive infinity, the polynomial goes to positive infinity, and as ( x ) approaches negative infinity, it goes to negative infinity. Since it's a cubic, it should cross the x-axis at least once. But since we have three real roots, it should cross three times.Given that we have three distinct roots, the graph should cross the x-axis at x=1, x=2, and x=4, each time crossing through, not touching and turning around, which would indicate a repeated root.Therefore, my conclusion is that the polynomial has three distinct real roots: 1, 2, and 4.So, the final answer is 1, 2, 4.

💡Alright, so I need to factor the expression (5y(y+2) + 8(y+2) + 15). Hmm, factoring expressions can sometimes be tricky, but let's take it step by step.First, I notice that both (5y(y+2)) and (8(y+2)) have a common factor of ((y+2)). That seems like a good starting point. Maybe I can factor out ((y+2)) from those two terms. Let me try that.So, if I factor out ((y+2)) from (5y(y+2) + 8(y+2)), I get:[(y+2)(5y + 8)]Okay, that part makes sense. Now, the original expression also has a constant term, (+15), which isn't multiplied by ((y+2)). So, after factoring out ((y+2)) from the first two terms, the expression becomes:[(y+2)(5y + 8) + 15]Now, I need to see if I can factor this further. The problem is that the (+15) isn't attached to ((y+2)), so it's just an extra term. I wonder if there's a way to combine this with the factored part.Let me think about it. If I expand ((y+2)(5y + 8)), what do I get? Let's multiply it out:[(y+2)(5y + 8) = y cdot 5y + y cdot 8 + 2 cdot 5y + 2 cdot 8 = 5y^2 + 8y + 10y + 16 = 5y^2 + 18y + 16]So, the expression becomes:[5y^2 + 18y + 16 + 15 = 5y^2 + 18y + 31]Hmm, now I have a quadratic expression (5y^2 + 18y + 31). I need to check if this can be factored further. To factor a quadratic of the form (ay^2 + by + c), I look for two numbers that multiply to (a cdot c) and add up to (b).In this case, (a = 5), (b = 18), and (c = 31). So, (a cdot c = 5 cdot 31 = 155). I need two numbers that multiply to 155 and add up to 18. Let's see:- 5 and 31: 5 + 31 = 36, which is too big.- 1 and 155: 1 + 155 = 156, way too big.- 5 and 31 are the only factors of 155 besides 1 and 155.Hmm, none of these pairs add up to 18. That means the quadratic (5y^2 + 18y + 31) doesn't factor nicely with integer coefficients. So, I can't factor this expression any further in a straightforward way.Wait a minute, maybe I made a mistake earlier. Let me go back to the original expression:[5y(y+2) + 8(y+2) + 15]I factored out ((y+2)) from the first two terms, getting ((y+2)(5y + 8) + 15). Maybe instead of stopping there, I can try to factor the entire expression differently.Is there a way to include the 15 with the ((y+2)) term? Let me think. If I consider the entire expression:[5y(y+2) + 8(y+2) + 15]I can write it as:[(5y + 8)(y + 2) + 15]But as I saw earlier, this doesn't help because 15 doesn't combine nicely with ((y+2)). Maybe I need to look for another common factor or try a different factoring technique.Another approach could be to see if the entire expression can be expressed as a product of two binomials. Let's assume it can be written as:[(ay + b)(cy + d)]Expanding this, we get:[acy^2 + (ad + bc)y + bd]Comparing this with our quadratic (5y^2 + 18y + 31), we have:- (ac = 5)- (ad + bc = 18)- (bd = 31)Since 5 is a prime number, the possible pairs for (a) and (c) are (1,5) or (5,1). Similarly, 31 is also a prime number, so the possible pairs for (b) and (d) are (1,31) or (31,1).Let's try (a = 5) and (c = 1). Then, we need to find (b) and (d) such that:- (bd = 31)- (ad + bc = 5d + b = 18)Trying (b = 1) and (d = 31):[5(31) + 1 = 155 + 1 = 156 neq 18]Not working.Trying (b = 31) and (d = 1):[5(1) + 31 = 5 + 31 = 36 neq 18]Still not working.Now, let's try (a = 1) and (c = 5). Then, we need:- (bd = 31)- (ad + bc = d + 5b = 18)Trying (b = 1) and (d = 31):[31 + 5(1) = 31 + 5 = 36 neq 18]Nope.Trying (b = 31) and (d = 1):[1 + 5(31) = 1 + 155 = 156 neq 18]Still not matching.It seems like no combination of these factors works. Therefore, the quadratic (5y^2 + 18y + 31) doesn't factor nicely with integer coefficients. This means that the expression can't be factored further beyond ((y+2)(5y + 8) + 15).Wait, but maybe I should check if there's a different way to factor the original expression without expanding it. Let's go back to:[5y(y+2) + 8(y+2) + 15]I factored out ((y+2)) from the first two terms, but perhaps there's another common factor I missed. Let me see.Looking at (5y(y+2) + 8(y+2)), I see that both terms have ((y+2)), so factoring that out is correct. The remaining part is (5y + 8), which doesn't have any common factors with 15. So, I don't think there's a way to factor out anything else.Alternatively, maybe I can rearrange the terms or group them differently. Let's try grouping the (5y(y+2)) and the (+15) together, and see if that helps:[5y(y+2) + 15 + 8(y+2)]Hmm, (5y(y+2) + 15) can be factored by taking out a 5:[5(y(y+2) + 3) + 8(y+2)]But that doesn't seem helpful because now I have:[5(y^2 + 2y + 3) + 8(y + 2)]Which expands to:[5y^2 + 10y + 15 + 8y + 16 = 5y^2 + 18y + 31]Again, back to the same quadratic that doesn't factor nicely.Maybe I should consider completing the square or using the quadratic formula, but since the problem asks for factoring, those methods might not be applicable here. Factoring usually refers to expressing the polynomial as a product of simpler polynomials with integer coefficients, which doesn't seem possible in this case.So, after trying different approaches and checking my work, it seems that the expression (5y(y+2) + 8(y+2) + 15) can be partially factored as ((y+2)(5y + 8) + 15), but it can't be factored further into simpler polynomials with integer coefficients. Therefore, the most factored form is:[(y+2)(5y + 8) + 15]I think that's as far as I can go with factoring this expression.

💡Okay, so I have this problem where I need to translate the function ( f(x) = sin(3x + frac{pi}{4}) ) by shifting its graph to the right by ( frac{pi}{3} ) units. After that, I need to find the minimum value of the resulting function ( y = g(x) ) on the interval ( [frac{pi}{3}, frac{2pi}{3}] ).Alright, let me start by recalling how function translations work. If I remember correctly, shifting a function to the right by ( c ) units involves replacing ( x ) with ( x - c ) in the function. So, for a general function ( f(x) ), shifting it right by ( c ) units gives ( f(x - c) ).Applying that to my function ( f(x) = sin(3x + frac{pi}{4}) ), shifting it right by ( frac{pi}{3} ) units should give me ( g(x) = sinleft(3(x - frac{pi}{3}) + frac{pi}{4}right) ). Let me write that out:[ g(x) = sinleft(3left(x - frac{pi}{3}right) + frac{pi}{4}right) ]Now, I need to simplify this expression. Let's distribute the 3 inside the sine function:[ g(x) = sinleft(3x - 3 cdot frac{pi}{3} + frac{pi}{4}right) ]Simplifying the constants:[ 3 cdot frac{pi}{3} = pi ]So,[ g(x) = sinleft(3x - pi + frac{pi}{4}right) ]Combine the constants:[ -pi + frac{pi}{4} = -frac{4pi}{4} + frac{pi}{4} = -frac{3pi}{4} ]Therefore, the function simplifies to:[ g(x) = sinleft(3x - frac{3pi}{4}right) ]Okay, so now I have the translated function ( g(x) = sinleft(3x - frac{3pi}{4}right) ). The next step is to find its minimum value on the interval ( [frac{pi}{3}, frac{2pi}{3}] ).To find the minimum value of a sine function, I know that the sine function oscillates between -1 and 1. So, the minimum value of ( sin(theta) ) is -1. However, I need to check if the argument ( 3x - frac{3pi}{4} ) reaches an angle where the sine function equals -1 within the given interval.Alternatively, I can analyze the behavior of ( g(x) ) over the interval by considering its derivative to find critical points, which might give me the minima or maxima.But before diving into calculus, maybe I can find the range of the argument ( 3x - frac{3pi}{4} ) when ( x ) is in ( [frac{pi}{3}, frac{2pi}{3}] ).Let me compute the argument at the endpoints:When ( x = frac{pi}{3} ):[ 3 cdot frac{pi}{3} - frac{3pi}{4} = pi - frac{3pi}{4} = frac{pi}{4} ]When ( x = frac{2pi}{3} ):[ 3 cdot frac{2pi}{3} - frac{3pi}{4} = 2pi - frac{3pi}{4} = frac{8pi}{4} - frac{3pi}{4} = frac{5pi}{4} ]So, the argument ( 3x - frac{3pi}{4} ) ranges from ( frac{pi}{4} ) to ( frac{5pi}{4} ) as ( x ) goes from ( frac{pi}{3} ) to ( frac{2pi}{3} ).Now, let me think about the sine function over the interval ( [frac{pi}{4}, frac{5pi}{4}] ). The sine function starts at ( sin(frac{pi}{4}) = frac{sqrt{2}}{2} ), goes up to 1 at ( frac{pi}{2} ), then back down to 0 at ( pi ), and continues down to ( -frac{sqrt{2}}{2} ) at ( frac{5pi}{4} ).Wait, but the minimum value of sine in this interval is actually at ( frac{5pi}{4} ), which is ( -frac{sqrt{2}}{2} ). However, I should check if the sine function reaches -1 within this interval. The sine function reaches -1 at ( frac{3pi}{2} ), which is approximately 4.712 radians. But ( frac{5pi}{4} ) is approximately 3.927 radians, which is less than ( frac{3pi}{2} ). So, the sine function doesn't reach -1 in this interval. Therefore, the minimum value is at ( frac{5pi}{4} ), which is ( -frac{sqrt{2}}{2} ).But let me verify this by considering the critical points. To find the extrema, I can take the derivative of ( g(x) ) and set it to zero.The derivative of ( g(x) = sin(3x - frac{3pi}{4}) ) is:[ g'(x) = 3cos(3x - frac{3pi}{4}) ]Setting the derivative equal to zero:[ 3cos(3x - frac{3pi}{4}) = 0 ][ cos(3x - frac{3pi}{4}) = 0 ]The solutions to ( cos(theta) = 0 ) are ( theta = frac{pi}{2} + kpi ) for integer ( k ).So,[ 3x - frac{3pi}{4} = frac{pi}{2} + kpi ][ 3x = frac{pi}{2} + kpi + frac{3pi}{4} ][ 3x = frac{2pi}{4} + frac{3pi}{4} + kpi ][ 3x = frac{5pi}{4} + kpi ][ x = frac{5pi}{12} + frac{kpi}{3} ]Now, let's find the values of ( x ) within ( [frac{pi}{3}, frac{2pi}{3}] ).For ( k = 0 ):[ x = frac{5pi}{12} approx 1.308 ]Which is approximately 1.308, and ( frac{pi}{3} approx 1.047 ), ( frac{2pi}{3} approx 2.094 ). So, ( frac{5pi}{12} ) is within the interval.For ( k = 1 ):[ x = frac{5pi}{12} + frac{pi}{3} = frac{5pi}{12} + frac{4pi}{12} = frac{9pi}{12} = frac{3pi}{4} approx 2.356 ]Which is greater than ( frac{2pi}{3} approx 2.094 ), so it's outside the interval.For ( k = -1 ):[ x = frac{5pi}{12} - frac{pi}{3} = frac{5pi}{12} - frac{4pi}{12} = frac{pi}{12} approx 0.261 ]Which is less than ( frac{pi}{3} approx 1.047 ), so also outside the interval.Therefore, the only critical point within the interval is at ( x = frac{5pi}{12} ).Now, let's evaluate ( g(x) ) at the critical point and at the endpoints to find the minimum.First, at ( x = frac{pi}{3} ):[ gleft(frac{pi}{3}right) = sinleft(3 cdot frac{pi}{3} - frac{3pi}{4}right) = sinleft(pi - frac{3pi}{4}right) = sinleft(frac{pi}{4}right) = frac{sqrt{2}}{2} ]At ( x = frac{5pi}{12} ):[ gleft(frac{5pi}{12}right) = sinleft(3 cdot frac{5pi}{12} - frac{3pi}{4}right) = sinleft(frac{15pi}{12} - frac{9pi}{12}right) = sinleft(frac{6pi}{12}right) = sinleft(frac{pi}{2}right) = 1 ]At ( x = frac{2pi}{3} ):[ gleft(frac{2pi}{3}right) = sinleft(3 cdot frac{2pi}{3} - frac{3pi}{4}right) = sinleft(2pi - frac{3pi}{4}right) = sinleft(frac{5pi}{4}right) = -frac{sqrt{2}}{2} ]So, evaluating ( g(x) ) at the critical point and endpoints, we get:- ( x = frac{pi}{3} ): ( frac{sqrt{2}}{2} )- ( x = frac{5pi}{12} ): ( 1 )- ( x = frac{2pi}{3} ): ( -frac{sqrt{2}}{2} )Therefore, the minimum value of ( g(x) ) on the interval ( [frac{pi}{3}, frac{2pi}{3}] ) is ( -frac{sqrt{2}}{2} ).I think that's it. I considered both the range of the argument and checked the critical points using calculus to ensure I didn't miss any minima or maxima within the interval. It looks like the minimum occurs at the right endpoint of the interval.

💡Alright, so I came across this interesting math problem: "Which numbers have the last two digits that match the last two digits of their square?" At first glance, it seems a bit tricky, but I think I can break it down step by step.Okay, let's start by understanding what the problem is asking. We need to find all numbers such that when you square them, the last two digits of the square are the same as the last two digits of the original number. In other words, if the number is, say, 25, then 25 squared is 625, and the last two digits of 625 are 25, which matches the original number. So, 25 is one such number.But how do we find all such numbers? I guess I need to think about how squaring affects the last two digits of a number. Maybe modular arithmetic can help here. Specifically, if we're interested in the last two digits, we're essentially working modulo 100.So, let's denote a number as ( n ). We want ( n^2 equiv n pmod{100} ). This simplifies to ( n^2 - n equiv 0 pmod{100} ), or ( n(n - 1) equiv 0 pmod{100} ). This means that the product of ( n ) and ( n - 1 ) must be divisible by 100.Now, 100 factors into ( 4 times 25 ), where 4 and 25 are coprime. So, by the Chinese Remainder Theorem, we can consider the problem modulo 4 and modulo 25 separately and then combine the results.First, let's consider ( n(n - 1) equiv 0 pmod{4} ). Since 4 is a power of 2, we need either ( n equiv 0 pmod{4} ) or ( n equiv 1 pmod{4} ).Next, consider ( n(n - 1) equiv 0 pmod{25} ). Here, 25 is a power of 5, so we need either ( n equiv 0 pmod{25} ) or ( n equiv 1 pmod{25} ).Now, we can combine these conditions using the Chinese Remainder Theorem. We have four cases to consider:1. ( n equiv 0 pmod{4} ) and ( n equiv 0 pmod{25} )2. ( n equiv 0 pmod{4} ) and ( n equiv 1 pmod{25} )3. ( n equiv 1 pmod{4} ) and ( n equiv 0 pmod{25} )4. ( n equiv 1 pmod{4} ) and ( n equiv 1 pmod{25} )Let's solve each case:1. **Case 1: ( n equiv 0 pmod{4} ) and ( n equiv 0 pmod{25} )** - This means ( n ) is divisible by both 4 and 25, so ( n equiv 0 pmod{100} ).2. **Case 2: ( n equiv 0 pmod{4} ) and ( n equiv 1 pmod{25} )** - We need to find a number that is 0 mod 4 and 1 mod 25. - Let ( n = 25k + 1 ). We need ( 25k + 1 equiv 0 pmod{4} ). - Since 25 mod 4 is 1, this becomes ( k + 1 equiv 0 pmod{4} ), so ( k equiv 3 pmod{4} ). - Therefore, ( n = 25(4m + 3) + 1 = 100m + 76 ). - So, ( n equiv 76 pmod{100} ).3. **Case 3: ( n equiv 1 pmod{4} ) and ( n equiv 0 pmod{25} )** - We need to find a number that is 1 mod 4 and 0 mod 25. - Let ( n = 25k ). We need ( 25k equiv 1 pmod{4} ). - Since 25 mod 4 is 1, this becomes ( k equiv 1 pmod{4} ). - Therefore, ( n = 25(4m + 1) = 100m + 25 ). - So, ( n equiv 25 pmod{100} ).4. **Case 4: ( n equiv 1 pmod{4} ) and ( n equiv 1 pmod{25} )** - This means ( n equiv 1 pmod{100} ).So, combining all four cases, the solutions modulo 100 are:- ( n equiv 0 pmod{100} )- ( n equiv 1 pmod{100} )- ( n equiv 25 pmod{100} )- ( n equiv 76 pmod{100} )Therefore, any number ending with 00, 01, 25, or 76 will have the property that the last two digits of its square match the last two digits of the number itself.To verify, let's test a few examples:- ( 0^2 = 0 ) → last two digits 00- ( 1^2 = 1 ) → last two digits 01- ( 25^2 = 625 ) → last two digits 25- ( 76^2 = 5776 ) → last two digits 76All of these satisfy the condition, so it seems like our solution is correct.In conclusion, the numbers that have the last two digits matching the last two digits of their square are those ending with 00, 01, 25, or 76.