answer：Okay, so I have this problem about a geometric sequence. Let me try to understand it step by step. The sequence is {a_n}, each term is positive, the first term is 1, and the common ratio is x. The sum of the first n terms is S_n. Then, they define f(x) as the limit as n approaches infinity of (2 S_n) divided by S_{n+1}. First, I need to find the analytical expression of f(x). Hmm, okay. So, let me recall what a geometric sequence is. It's a sequence where each term is the previous term multiplied by a common ratio, which in this case is x. Since the first term is 1, the sequence is 1, x, x^2, x^3, and so on. Now, the sum of the first n terms of a geometric sequence is given by S_n = (1 - x^n)/(1 - x) when x ≠ 1. If x = 1, then it's just S_n = n, because each term is 1. So, I need to consider different cases for x because the behavior of the sequence changes depending on whether x is less than 1, equal to 1, or greater than 1.Let me start with x = 1. If x is 1, then S_n is just n, as I mentioned. So, S_{n+1} would be n + 1. Then, f(x) is the limit as n approaches infinity of (2n)/(n + 1). Simplifying that, I can divide numerator and denominator by n, which gives 2/(1 + 1/n). As n approaches infinity, 1/n approaches 0, so the limit is 2. So, f(1) = 2.Next, let's consider x > 1. In this case, the terms of the sequence are increasing because each term is x times the previous one, and x is greater than 1. So, as n increases, x^n becomes very large. The sum S_n is (1 - x^n)/(1 - x). Similarly, S_{n+1} is (1 - x^{n+1})/(1 - x). So, f(x) is the limit as n approaches infinity of [2*(1 - x^n)/(1 - x)] divided by [(1 - x^{n+1})/(1 - x)]. The (1 - x) terms cancel out, so we're left with 2*(1 - x^n)/(1 - x^{n+1}).Now, as n approaches infinity, x^n and x^{n+1} both approach infinity because x > 1. So, 1 - x^n is approximately -x^n, and 1 - x^{n+1} is approximately -x^{n+1}. So, the ratio becomes 2*(-x^n)/(-x^{n+1}) = 2*(x^n)/(x^{n+1}) = 2/x. Wait, that's different from what I thought earlier. Did I make a mistake?Wait, let me check again. If x > 1, then x^n and x^{n+1} go to infinity, so 1 - x^n ≈ -x^n, and 1 - x^{n+1} ≈ -x^{n+1}. So, the ratio is 2*(-x^n)/(-x^{n+1}) = 2*(x^n)/(x^{n+1}) = 2/x. So, f(x) = 2/x for x > 1. But wait, when I plug in x = 1, 2/x would be 2, which matches the previous case. So, maybe f(x) = 2/x for x ≥ 1.But wait, earlier I thought it was 2, but now I'm getting 2/x. Hmm, maybe I made a mistake in the initial assumption. Let me think again.When x > 1, S_n = (1 - x^n)/(1 - x). As n approaches infinity, x^n approaches infinity, so S_n approaches infinity. Similarly, S_{n+1} also approaches infinity. So, the ratio S_n/S_{n+1} is (1 - x^n)/(1 - x^{n+1}). Let's factor out x^n from numerator and denominator: (x^n*(1/x^n - 1))/(x^{n+1}*(1/x^{n+1} - 1)). Simplifying, that's (x^n*(1 - x^n)/x^n)/(x^{n+1}*(1 - x^{n+1})/x^{n+1})) = (1 - x^n)/(1 - x^{n+1}). Wait, that doesn't help much.Alternatively, let's divide numerator and denominator by x^n: (1/x^n - 1)/(1/x^n - x). As n approaches infinity, 1/x^n approaches 0, so the ratio becomes (-1)/(-x) = 1/x. Therefore, S_n/S_{n+1} approaches 1/x. So, f(x) = 2*(1/x) = 2/x.Wait, so earlier I thought f(x) was 2 for x > 1, but now I'm getting 2/x. Which one is correct? Let me test with x = 2. If x = 2, then S_n = (1 - 2^n)/(1 - 2) = (1 - 2^n)/(-1) = 2^n - 1. Similarly, S_{n+1} = 2^{n+1} - 1. So, f(x) = lim_{n→∞} [2*(2^n - 1)]/(2^{n+1} - 1). Simplify numerator: 2^{n+1} - 2. Denominator: 2^{n+1} - 1. So, the ratio is (2^{n+1} - 2)/(2^{n+1} - 1). As n approaches infinity, both numerator and denominator approach 2^{n+1}, so the limit is 1. Therefore, f(2) = 1, which is 2/2 = 1. So, that matches 2/x. So, f(x) = 2/x for x > 1.Wait, but earlier when I thought x = 1, f(x) = 2, which is also 2/1 = 2. So, maybe f(x) = 2/x for x ≥ 1. Okay, that makes sense.Now, let's consider 0 < x < 1. In this case, the terms of the sequence are decreasing because each term is x times the previous one, and x is less than 1. So, as n increases, x^n approaches 0. The sum S_n is (1 - x^n)/(1 - x). Similarly, S_{n+1} is (1 - x^{n+1})/(1 - x). So, f(x) is the limit as n approaches infinity of [2*(1 - x^n)/(1 - x)] divided by [(1 - x^{n+1})/(1 - x)]. Again, the (1 - x) terms cancel out, so we're left with 2*(1 - x^n)/(1 - x^{n+1}).As n approaches infinity, x^n approaches 0 and x^{n+1} also approaches 0. So, the numerator becomes 2*(1 - 0) = 2, and the denominator becomes 1 - 0 = 1. So, f(x) = 2/1 = 2. Wait, that can't be right because when x approaches 1 from below, f(x) should approach 2 as well, but for x < 1, it's also 2? That doesn't seem right.Wait, let me check with x = 1/2. If x = 1/2, then S_n = (1 - (1/2)^n)/(1 - 1/2) = (1 - (1/2)^n)/(1/2) = 2*(1 - (1/2)^n). Similarly, S_{n+1} = 2*(1 - (1/2)^{n+1}). So, f(x) = lim_{n→∞} [2*2*(1 - (1/2)^n)]/[2*(1 - (1/2)^{n+1})] = lim_{n→∞} [4*(1 - (1/2)^n)]/[2*(1 - (1/2)^{n+1})] = lim_{n→∞} [2*(1 - (1/2)^n)]/[1 - (1/2)^{n+1}].As n approaches infinity, (1/2)^n approaches 0, so the ratio becomes 2*(1 - 0)/(1 - 0) = 2. So, f(1/2) = 2. Hmm, but that's the same as when x = 1. That seems odd because when x approaches 1 from below, f(x) should approach 2, but for x < 1, it's also 2. Wait, that can't be right because when x approaches 0, f(x) should approach something else.Wait, let me think again. If x approaches 0, then S_n = (1 - x^n)/(1 - x) ≈ 1/(1 - x) because x^n is negligible. Similarly, S_{n+1} ≈ 1/(1 - x). So, f(x) = lim_{n→∞} [2*S_n]/S_{n+1} ≈ [2*(1/(1 - x))]/(1/(1 - x)) = 2. So, even as x approaches 0, f(x) approaches 2. That seems strange because I thought maybe it would approach something else.Wait, but let's test with x = 1/2 again. We saw that f(x) = 2. So, maybe for all 0 < x < 1, f(x) = 2. But that contradicts my earlier thought that it might be different. Wait, let me check with x = 1/3.If x = 1/3, then S_n = (1 - (1/3)^n)/(1 - 1/3) = (1 - (1/3)^n)/(2/3) = (3/2)*(1 - (1/3)^n). Similarly, S_{n+1} = (3/2)*(1 - (1/3)^{n+1}). So, f(x) = lim_{n→∞} [2*(3/2)*(1 - (1/3)^n)] / [ (3/2)*(1 - (1/3)^{n+1}) ] = lim_{n→∞} [3*(1 - (1/3)^n)] / [ (3/2)*(1 - (1/3)^{n+1}) ] = lim_{n→∞} [2*(1 - (1/3)^n)] / [1 - (1/3)^{n+1}].As n approaches infinity, (1/3)^n approaches 0, so the ratio becomes 2*(1 - 0)/(1 - 0) = 2. So, f(1/3) = 2. Hmm, so it seems that for any x in (0,1), f(x) = 2. But that contradicts my initial thought that it might be different. Wait, maybe I'm missing something.Wait, let me think about the general case for 0 < x < 1. S_n = (1 - x^n)/(1 - x). So, f(x) = lim_{n→∞} [2*(1 - x^n)/(1 - x)] / [ (1 - x^{n+1})/(1 - x) ] = lim_{n→∞} 2*(1 - x^n)/(1 - x^{n+1}).As n approaches infinity, x^n approaches 0, so 1 - x^n approaches 1, and 1 - x^{n+1} approaches 1. Therefore, f(x) = 2*1/1 = 2. So, for any 0 < x < 1, f(x) = 2.Wait, but that seems counterintuitive because when x is very small, say x approaches 0, the sum S_n approaches 1/(1 - x) which is approximately 1. So, S_n is approximately 1, and S_{n+1} is also approximately 1. So, the ratio 2*S_n/S_{n+1} is approximately 2*1/1 = 2. So, that makes sense.But wait, when x is close to 1, say x = 0.99, then S_n = (1 - 0.99^n)/(1 - 0.99). As n increases, 0.99^n approaches 0, so S_n approaches 1/(1 - 0.99) = 10. Similarly, S_{n+1} approaches 10. So, f(x) = 2*10/10 = 2. So, even when x is close to 1, f(x) is 2.Wait, but earlier when x > 1, f(x) = 2/x. So, for x = 2, f(x) = 1, for x = 3, f(x) = 2/3, and so on. So, the function f(x) is 2/x for x ≥ 1 and 2 for 0 < x < 1.But wait, when x = 1, f(x) = 2, which is consistent with both cases because 2/x when x = 1 is 2, and for 0 < x < 1, it's also 2. So, the function is piecewise defined as 2 for 0 < x ≤ 1 and 2/x for x > 1.Wait, but in the case when x = 1, the sum S_n = n, so f(x) = lim_{n→∞} 2n/(n+1) = 2. So, that's consistent.So, putting it all together, f(x) is:- 2 for 0 < x ≤ 1- 2/x for x > 1Now, for part 2, plotting the graph of y = f(x). Let's think about how this function behaves.For 0 < x ≤ 1, y = 2 is a constant function. So, it's a horizontal line at y = 2 from x = 0 to x = 1.For x > 1, y = 2/x is a hyperbola that decreases as x increases. At x = 1, y = 2, and as x approaches infinity, y approaches 0.So, the graph would have a horizontal line at y = 2 from x = 0 to x = 1, and then a hyperbola starting at (1, 2) and decreasing towards the x-axis as x increases.I should also note that at x = 1, both definitions of f(x) give y = 2, so the function is continuous at x = 1.Wait, but let me double-check the behavior as x approaches 1 from below and above. As x approaches 1 from below, y approaches 2. As x approaches 1 from above, y approaches 2 as well because 2/x approaches 2. So, the function is continuous at x = 1.Therefore, the graph is a horizontal line at y = 2 from x = 0 to x = 1, and a hyperbola y = 2/x for x > 1.I think that's the correct analysis. So, summarizing:1. f(x) is 2 for 0 < x ≤ 1 and 2/x for x > 1.2. The graph consists of a horizontal line at y = 2 from x = 0 to x = 1 and a hyperbola y = 2/x for x > 1.

answer：Alright, so I've got this logic puzzle here about three gnomes named Pili, Eli, and Spali. They found some treasures in a cave: a diamond, a topaz, and a copper basin. The goal is to figure out who found what based on the clues provided. Let me try to break this down step by step.First, let's list out the clues:1. Eli has a red hood and a beard longer than Pili’s.2. The one who found the basin has the longest beard and a blue hood.3. The gnome with the shortest beard found the diamond.Okay, so we have three gnomes, each with a different beard length and a different treasure. Also, their hoods are either red or blue. Let's see how we can use these clues to figure out who found what.Starting with the first clue: Eli has a red hood and a beard longer than Pili’s. So, Eli's beard is longer than Pili's. That means Pili has a shorter beard than Eli. But we don't know yet how Spali's beard compares to the others.The second clue says that the one who found the basin has the longest beard and a blue hood. So, the basin finder has the longest beard and a blue hood. Since Eli has a red hood, Eli cannot be the one who found the basin. That leaves Pili and Spali as possibilities for finding the basin.The third clue tells us that the gnome with the shortest beard found the diamond. So, whoever has the shortest beard is the one who found the diamond.Let me try to organize this information.From clue 1:- Eli's beard > Pili's beardFrom clue 2:- Basin finder has the longest beard and a blue hood.From clue 3:- Shortest beard found the diamond.So, let's consider the beard lengths. We have three gnomes: Pili, Eli, and Spali. Their beards can be ordered from shortest to longest.From clue 1, we know that Eli's beard is longer than Pili's. So, Pili < Eli in beard length.Now, we don't know where Spali fits in. Spali could have the longest beard, or Eli could have the longest beard. But clue 2 says that the basin finder has the longest beard. Also, the basin finder has a blue hood.Eli has a red hood, so he can't be the basin finder. Therefore, the basin finder must be either Pili or Spali, and they must have a blue hood.But clue 2 says the basin finder has the longest beard. So, if Spali has the longest beard, then Spali found the basin. If Eli had the longest beard, but he can't be the basin finder because he has a red hood, so Spali must have the longest beard.Therefore, Spali has the longest beard and a blue hood, so he found the basin.Now, from clue 1, we know that Eli's beard is longer than Pili's, and Spali has the longest beard. So, the order of beard lengths is:Pili < Eli < SpaliSo, Pili has the shortest beard, Eli has the middle-length beard, and Spali has the longest beard.From clue 3, the gnome with the shortest beard found the diamond. Since Pili has the shortest beard, Pili found the diamond.Now, the remaining treasure is the topaz. The only gnome left is Eli, so Eli must have found the topaz.Let me double-check to make sure this makes sense.- Spali has the longest beard and a blue hood, so he found the basin.- Pili has the shortest beard, so he found the diamond.- Eli has the middle-length beard and a red hood, so he found the topaz.Yes, that seems to fit all the clues.So, to summarize:- Pili found the diamond.- Eli found the topaz.- Spali found the copper basin.

answer：Alright, so I've got this problem here with four different operations labeled A, B, C, and D. I need to figure out which ones are correct. Let me take it step by step.Starting with option A: ( a^{6} div a^{-2} = a^{4} ). Hmm, okay, I remember that when you divide exponents with the same base, you subtract the exponents. So, ( a^{6} div a^{-2} ) should be ( a^{6 - (-2)} ). Wait, subtracting a negative is like adding, so that would be ( a^{6 + 2} = a^{8} ). But the option says it's ( a^{4} ). That doesn't seem right. Maybe I did something wrong? Let me double-check. No, I think I'm correct. Dividing ( a^{6} ) by ( a^{-2} ) should increase the exponent, not decrease it. So, A is probably incorrect.Moving on to option B: ( (-2t) cdot (3t + t^{2} - 1) = -6t^{2} - 2t^{3} + 2t ). Okay, let's distribute the ( -2t ) across each term inside the parentheses. First term: ( -2t cdot 3t = -6t^{2} ). That matches the given answer.Second term: ( -2t cdot t^{2} = -2t^{3} ). That also matches.Third term: ( -2t cdot (-1) = 2t ). Yep, that's correct too.So, all three terms when multiplied out give the same result as in option B. So, B seems correct.Option C: ( (-2xy^{3})^{2} = 4x^{2}y^{6} ). Alright, when you raise a product to a power, you raise each factor to that power. So, ( (-2)^{2} = 4 ), ( x^{2} ) stays as ( x^{2} ), and ( (y^{3})^{2} = y^{6} ). Multiplying those together gives ( 4x^{2}y^{6} ), which matches the given answer. So, C is correct.Lastly, option D: ( (a - b) cdot (a + b) = a^{2} - bc ). Hmm, I recall that ( (a - b)(a + b) ) is a difference of squares, which should be ( a^{2} - b^{2} ). But here it's saying ( a^{2} - bc ). That doesn't look right. Maybe it's a typo or something? Unless there's more context, but as it stands, it should be ( a^{2} - b^{2} ), not ( a^{2} - bc ). So, D is incorrect.So, after checking all four options, it looks like B and C are the correct ones.

answer：Okay, so I have this function ( f(x) = 2sin^{2}left(frac{pi}{4} + xright) - sqrt{3}cos 2x ) and I need to find its minimum value on the interval ( frac{pi}{4} leq x leq frac{pi}{2} ). Hmm, let me try to figure this out step by step.First, I remember that ( sin^2 theta ) can be rewritten using the double-angle identity. The identity is ( sin^2 theta = frac{1 - cos 2theta}{2} ). So, maybe I can apply that here to simplify the function.Let me rewrite the ( 2sin^{2}left(frac{pi}{4} + xright) ) part:( 2sin^{2}left(frac{pi}{4} + xright) = 2 times frac{1 - cosleft(2left(frac{pi}{4} + xright)right)}{2} )Simplifying that, the 2 and the denominator 2 cancel out, so:( 1 - cosleft(frac{pi}{2} + 2xright) )Wait, ( cosleft(frac{pi}{2} + 2xright) ) is another trigonometric identity. I think ( cosleft(frac{pi}{2} + thetaright) = -sin theta ). Let me confirm that. Yes, because cosine of an angle plus pi/2 is the negative sine of that angle.So, substituting that in:( 1 - (-sin 2x) = 1 + sin 2x )Okay, so now the function becomes:( f(x) = 1 + sin 2x - sqrt{3}cos 2x )Hmm, so now I have ( f(x) = 1 + sin 2x - sqrt{3}cos 2x ). That looks like a combination of sine and cosine terms. Maybe I can combine them into a single sine or cosine function using the amplitude-phase form.I recall that ( Asin theta + Bcos theta = Csin(theta + phi) ) or ( Ccos(theta + phi) ), where ( C = sqrt{A^2 + B^2} ) and ( phi ) is the phase shift. Let me try that.In this case, the coefficients are ( A = 1 ) for sine and ( B = -sqrt{3} ) for cosine. So, the amplitude ( C ) would be:( C = sqrt{1^2 + (-sqrt{3})^2} = sqrt{1 + 3} = sqrt{4} = 2 )Okay, so the amplitude is 2. Now, to find the phase shift ( phi ), I can use:( tan phi = frac{B}{A} = frac{-sqrt{3}}{1} = -sqrt{3} )So, ( phi = arctan(-sqrt{3}) ). Hmm, arctangent of negative sqrt(3). I know that ( tan(frac{pi}{3}) = sqrt{3} ), so ( tan(-frac{pi}{3}) = -sqrt{3} ). Therefore, ( phi = -frac{pi}{3} ).So, putting it all together, the expression ( sin 2x - sqrt{3}cos 2x ) can be written as:( 2sinleft(2x - frac{pi}{3}right) )Therefore, the function ( f(x) ) becomes:( f(x) = 1 + 2sinleft(2x - frac{pi}{3}right) )Alright, so now I have ( f(x) = 1 + 2sinleft(2x - frac{pi}{3}right) ). To find the minimum value of this function on the interval ( frac{pi}{4} leq x leq frac{pi}{2} ), I need to analyze the sine function.The sine function oscillates between -1 and 1, so ( sinleft(2x - frac{pi}{3}right) ) will also oscillate between -1 and 1. Therefore, ( 2sinleft(2x - frac{pi}{3}right) ) will oscillate between -2 and 2, and adding 1 will shift it to oscillate between -1 and 3.But I need to find the minimum value specifically on the interval ( frac{pi}{4} leq x leq frac{pi}{2} ). So, I should consider the range of ( 2x - frac{pi}{3} ) as x varies from ( frac{pi}{4} ) to ( frac{pi}{2} ).Let me compute the bounds for ( 2x - frac{pi}{3} ):When ( x = frac{pi}{4} ):( 2 times frac{pi}{4} - frac{pi}{3} = frac{pi}{2} - frac{pi}{3} = frac{3pi}{6} - frac{2pi}{6} = frac{pi}{6} )When ( x = frac{pi}{2} ):( 2 times frac{pi}{2} - frac{pi}{3} = pi - frac{pi}{3} = frac{3pi}{3} - frac{pi}{3} = frac{2pi}{3} )So, ( 2x - frac{pi}{3} ) ranges from ( frac{pi}{6} ) to ( frac{2pi}{3} ) as x goes from ( frac{pi}{4} ) to ( frac{pi}{2} ).Now, let's consider the sine function over the interval ( frac{pi}{6} ) to ( frac{2pi}{3} ). The sine function increases from ( frac{pi}{6} ) to ( frac{pi}{2} ), reaching its maximum at ( frac{pi}{2} ), and then decreases from ( frac{pi}{2} ) to ( frac{2pi}{3} ).So, the minimum value of ( sintheta ) on ( frac{pi}{6} leq theta leq frac{2pi}{3} ) occurs at the endpoints. Let me evaluate ( sintheta ) at ( theta = frac{pi}{6} ) and ( theta = frac{2pi}{3} ):At ( theta = frac{pi}{6} ):( sinleft(frac{pi}{6}right) = frac{1}{2} )At ( theta = frac{2pi}{3} ):( sinleft(frac{2pi}{3}right) = sinleft(pi - frac{pi}{3}right) = sinleft(frac{pi}{3}right) = frac{sqrt{3}}{2} )Wait, so both endpoints give positive values. But since the sine function reaches a maximum at ( frac{pi}{2} ) and is symmetric around ( frac{pi}{2} ), the minimum on this interval is actually at the lower endpoint, which is ( frac{pi}{6} ), giving ( frac{1}{2} ).But hold on, is that correct? Because ( sintheta ) is increasing from ( frac{pi}{6} ) to ( frac{pi}{2} ) and decreasing from ( frac{pi}{2} ) to ( frac{2pi}{3} ). So, the minimum should be at the lower end, ( frac{pi}{6} ), and the maximum at ( frac{pi}{2} ).Therefore, the minimum value of ( sintheta ) in this interval is ( frac{1}{2} ), and the maximum is 1.But wait, if I plug ( theta = frac{pi}{6} ) into ( 2sintheta ), I get ( 2 times frac{1}{2} = 1 ), and adding 1 gives ( f(x) = 2 ).Similarly, at ( theta = frac{2pi}{3} ), ( 2sinleft(frac{2pi}{3}right) = 2 times frac{sqrt{3}}{2} = sqrt{3} ), and adding 1 gives ( f(x) = 1 + sqrt{3} approx 2.732 ).At ( theta = frac{pi}{2} ), ( 2sinleft(frac{pi}{2}right) = 2 times 1 = 2 ), and adding 1 gives ( f(x) = 3 ).Wait, so the function reaches a maximum of 3 at ( x = frac{pi}{2} ), but the minimum seems to be at ( x = frac{pi}{4} ), giving ( f(x) = 2 ).But hold on, is that the actual minimum? Let me double-check.The function ( f(x) = 1 + 2sinleft(2x - frac{pi}{3}right) ) will have its minimum when ( sinleft(2x - frac{pi}{3}right) ) is minimized. Since the sine function can go as low as -1, but in our interval ( frac{pi}{6} leq 2x - frac{pi}{3} leq frac{2pi}{3} ), the sine function doesn't reach -1. The lowest it goes is ( frac{1}{2} ) at ( frac{pi}{6} ).But wait, actually, in the interval ( frac{pi}{6} ) to ( frac{2pi}{3} ), the sine function starts at ( frac{1}{2} ), goes up to 1 at ( frac{pi}{2} ), and then back down to ( frac{sqrt{3}}{2} ) at ( frac{2pi}{3} ). So, the minimum value in this interval is indeed ( frac{1}{2} ), which occurs at ( theta = frac{pi}{6} ) or ( x = frac{pi}{4} ).Therefore, the minimum value of ( f(x) ) is ( 1 + 2 times frac{1}{2} = 1 + 1 = 2 ).But just to be thorough, let me check if there's any critical point inside the interval where the function might attain a lower value. To do that, I can take the derivative of ( f(x) ) and set it to zero.Given ( f(x) = 1 + 2sinleft(2x - frac{pi}{3}right) ), the derivative is:( f'(x) = 2 times cosleft(2x - frac{pi}{3}right) times 2 = 4cosleft(2x - frac{pi}{3}right) )Setting the derivative equal to zero:( 4cosleft(2x - frac{pi}{3}right) = 0 )Which simplifies to:( cosleft(2x - frac{pi}{3}right) = 0 )The solutions to this equation are:( 2x - frac{pi}{3} = frac{pi}{2} + kpi ), where ( k ) is an integer.Solving for x:( 2x = frac{pi}{2} + frac{pi}{3} + kpi = frac{3pi}{6} + frac{2pi}{6} + kpi = frac{5pi}{6} + kpi )So,( x = frac{5pi}{12} + frac{kpi}{2} )Now, let's see if any of these solutions lie within our interval ( frac{pi}{4} leq x leq frac{pi}{2} ).For ( k = 0 ):( x = frac{5pi}{12} approx 1.308 ) radians.Since ( frac{pi}{4} approx 0.785 ) and ( frac{pi}{2} approx 1.571 ), ( frac{5pi}{12} ) is approximately 1.308, which is within the interval.For ( k = 1 ):( x = frac{5pi}{12} + frac{pi}{2} = frac{5pi}{12} + frac{6pi}{12} = frac{11pi}{12} approx 2.879 ), which is outside the interval.Similarly, negative k would give x less than ( frac{pi}{4} ), which is also outside the interval.So, the only critical point inside the interval is at ( x = frac{5pi}{12} ).Let me compute the value of ( f(x) ) at this critical point.First, compute ( 2x - frac{pi}{3} ):( 2 times frac{5pi}{12} - frac{pi}{3} = frac{10pi}{12} - frac{4pi}{12} = frac{6pi}{12} = frac{pi}{2} )So,( fleft(frac{5pi}{12}right) = 1 + 2sinleft(frac{pi}{2}right) = 1 + 2 times 1 = 3 )So, at ( x = frac{5pi}{12} ), the function reaches a maximum of 3.Therefore, the critical point inside the interval is a maximum, not a minimum. So, the minimum must occur at one of the endpoints.Earlier, I found that at ( x = frac{pi}{4} ), ( f(x) = 2 ), and at ( x = frac{pi}{2} ), ( f(x) = 1 + sqrt{3} approx 2.732 ). So, the minimum value is indeed 2 at ( x = frac{pi}{4} ).Just to make sure, let me compute ( f(x) ) at ( x = frac{pi}{4} ):( fleft(frac{pi}{4}right) = 2sin^{2}left(frac{pi}{4} + frac{pi}{4}right) - sqrt{3}cosleft(2 times frac{pi}{4}right) )Simplify inside the sine:( frac{pi}{4} + frac{pi}{4} = frac{pi}{2} )So,( 2sin^{2}left(frac{pi}{2}right) = 2 times 1^2 = 2 )And,( cosleft(frac{pi}{2}right) = 0 )So,( fleft(frac{pi}{4}right) = 2 - sqrt{3} times 0 = 2 )Yep, that's correct.And at ( x = frac{pi}{2} ):( fleft(frac{pi}{2}right) = 2sin^{2}left(frac{pi}{4} + frac{pi}{2}right) - sqrt{3}cosleft(2 times frac{pi}{2}right) )Simplify inside the sine:( frac{pi}{4} + frac{pi}{2} = frac{3pi}{4} )So,( 2sin^{2}left(frac{3pi}{4}right) = 2 times left(frac{sqrt{2}}{2}right)^2 = 2 times frac{1}{2} = 1 )And,( cosleft(piright) = -1 )So,( fleft(frac{pi}{2}right) = 1 - sqrt{3} times (-1) = 1 + sqrt{3} approx 2.732 )Which confirms the earlier calculation.Therefore, after checking both endpoints and the critical point inside the interval, the minimum value of the function ( f(x) ) on ( frac{pi}{4} leq x leq frac{pi}{2} ) is indeed 2.**Final Answer**The minimum value of the function is boxed{2}.

answer：Okay, so I have this problem here: If ( x = 2 ), then ( x^2 - 4 = 0 ). How many of the following statements are true? Hmm, I need to figure out how many of these statements are actually true. Let me start by understanding what each statement is saying.First, the original statement is: If ( x = 2 ), then ( x^2 - 4 = 0 ). That makes sense because if I plug in 2 for x, I get ( 2^2 - 4 = 4 - 4 = 0 ). So, the original statement is definitely true.Now, I think there are related statements like the converse, inverse, and contrapositive. Let me recall what those are. The converse of a statement switches the hypothesis and the conclusion. So, the converse would be: If ( x^2 - 4 = 0 ), then ( x = 2 ). Hmm, is that true? Well, ( x^2 - 4 = 0 ) can be factored into ( (x - 2)(x + 2) = 0 ), which means ( x = 2 ) or ( x = -2 ). So, just because ( x^2 - 4 = 0 ) doesn't necessarily mean ( x = 2 ); it could also be ( x = -2 ). Therefore, the converse is not always true. So, the converse is false.Next, the inverse statement. The inverse negates both the hypothesis and the conclusion. So, the inverse would be: If ( x neq 2 ), then ( x^2 - 4 neq 0 ). Is that true? Well, if ( x ) is not 2, does that mean ( x^2 - 4 ) can't be zero? But wait, if ( x = -2 ), which is not equal to 2, then ( x^2 - 4 = (-2)^2 - 4 = 4 - 4 = 0 ). So, even though ( x ) is not 2, ( x^2 - 4 ) can still be zero. Therefore, the inverse is also false.Lastly, the contrapositive. The contrapositive negates both the hypothesis and the conclusion and switches them. So, the contrapositive would be: If ( x^2 - 4 neq 0 ), then ( x neq 2 ). Is that true? Well, if ( x^2 - 4 ) is not zero, that means ( x ) is neither 2 nor -2. So, yes, if ( x^2 - 4 neq 0 ), then ( x ) cannot be 2. Therefore, the contrapositive is true.So, summarizing:1. Original statement: True2. Converse: False3. Inverse: False4. Contrapositive: TrueThat means out of the four statements, two of them are true. I think that's the answer.

answer：To find the largest four-digit number divisible by 33, we start with the largest four-digit number, which is 9999. We then check if 9999 is divisible by 33.First, we divide 9999 by 33:[9999 div 33 = 303 text{ with a remainder of } 0]Since the remainder is 0, 9999 is divisible by 33. Therefore, 9999 is the largest four-digit number that is divisible by 33.