💡Alright, let's tackle this problem step by step. So, we have a group of students that starts by standing in rows with 20 students each on June 1. Then, from June 2 to June 15, they rearrange themselves into a different number of rows each day. On June 16, they try to find a new configuration but fail. Additionally, they notice that they couldn't form rows of 10, 25, or 50 students during these days. We need to find the smallest possible number of students in the group.First, let's understand what the problem is asking. The number of students must be such that it can be divided into different numbers of rows each day from June 2 to June 15. That means the total number of students must have at least 15 different divisors (since each divisor represents a possible number of rows). However, on June 16, they couldn't find a new configuration, implying that they've exhausted all possible divisors by then. Also, they couldn't form rows of 10, 25, or 50 students, which means these numbers are not divisors of the total number of students.So, to rephrase, we need a number that has exactly 15 divisors, excluding 10, 25, and 50. Moreover, since on June 1, they were standing in rows of 20, the total number of students must be divisible by 20.Let's recall that the number of divisors of a number can be determined by its prime factorization. If a number N can be expressed as ( N = p_1^{a_1} cdot p_2^{a_2} cdot ldots cdot p_n^{a_n} ), then the number of divisors is ( (a_1 + 1)(a_2 + 1) ldots (a_n + 1) ).Given that the number must be divisible by 20, which factors into ( 2^2 cdot 5^1 ), the prime factors of N must include at least ( 2^2 ) and ( 5^1 ).Now, we need N to have exactly 15 divisors. Let's find the possible exponents in the prime factorization that would result in 15 divisors. Since 15 factors into 15 = 15 × 1 or 15 = 5 × 3. Therefore, the exponents in the prime factorization could be:1. ( (14) ) because ( 14 + 1 = 15 )2. ( (4, 2) ) because ( (4 + 1)(2 + 1) = 5 × 3 = 15 )So, the possible prime factorizations are either a single prime raised to the 14th power or two primes where one is raised to the 4th power and the other to the 2nd power.However, since N must be divisible by 20 (( 2^2 cdot 5^1 )), the prime factors must include at least 2 and 5. Therefore, the second option (( (4, 2) )) is more plausible because it allows for multiple prime factors.Let's consider the second option: ( N = p_1^4 cdot p_2^2 ). To minimize N, we should choose the smallest primes, which are 2 and 5. So, let's assign ( p_1 = 2 ) and ( p_2 = 5 ). Then, ( N = 2^4 cdot 5^2 = 16 cdot 25 = 400 ). But 400 is quite large, and we might be able to find a smaller N by adjusting the exponents or choosing different primes.Wait, perhaps we can have more than two prime factors. Let's see. If we have three primes, say 2, 3, and 5, then the number of divisors would be ( (a+1)(b+1)(c+1) ). We need this product to be 15. Since 15 = 3 × 5, we can have exponents such that ( (a+1)(b+1)(c+1) = 3 × 5 ). This could be achieved by exponents (2,4,0), but since exponents can't be negative, we need to adjust.Alternatively, if we have exponents (2,1,1), then the number of divisors would be ( (2+1)(1+1)(1+1) = 3 × 2 × 2 = 12 ), which is less than 15. So, that doesn't work.What if we have exponents (4,2,0)? Then, the number of divisors would be ( (4+1)(2+1)(0+1) = 5 × 3 × 1 = 15 ). That works. So, N could be ( 2^4 cdot 3^2 cdot 5^0 = 16 × 9 = 144 ). But 144 is still larger than some of the options given.Wait, let's check the options provided: A) 90, B) 100, C) 110, D) 120, E) 150.We need to find the smallest N among these options that satisfies the conditions.Let's check each option:A) 90: Prime factorization is ( 2 × 3^2 × 5 ). Number of divisors is ( (1+1)(2+1)(1+1) = 2 × 3 × 2 = 12 ). But we need 15 divisors, so this doesn't work.B) 100: Prime factorization is ( 2^2 × 5^2 ). Number of divisors is ( (2+1)(2+1) = 9 ). Not enough.C) 110: Prime factorization is ( 2 × 5 × 11 ). Number of divisors is ( (1+1)(1+1)(1+1) = 8 ). Not enough.D) 120: Prime factorization is ( 2^3 × 3 × 5 ). Number of divisors is ( (3+1)(1+1)(1+1) = 4 × 2 × 2 = 16 ). Hmm, that's more than 15. But wait, the problem says they couldn't form rows of 10, 25, or 50. So, we need to exclude these divisors.Let's list the divisors of 120:1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.So, there are 16 divisors. But they couldn't form rows of 10, 25, or 50. However, 25 and 50 are not divisors of 120, so we only need to exclude 10. That leaves us with 15 valid divisors. Perfect!E) 150: Prime factorization is ( 2 × 3 × 5^2 ). Number of divisors is ( (1+1)(1+1)(2+1) = 2 × 2 × 3 = 12 ). Not enough.So, the smallest number that fits all the conditions is 120.

💡First, I recognize that in a parallelogram, opposite angles are equal. This means that angle ( A ) is equal to angle ( C ).Given that ( angle A + angle C = 110^circ ), and since ( angle A = angle C ), I can set up the equation:[2angle A = 110^circ]Solving for ( angle A ) gives:[angle A = frac{110^circ}{2} = 55^circ]Therefore, ( angle C = 55^circ ) as well.Next, I know that consecutive angles in a parallelogram are supplementary, meaning they add up to ( 180^circ ). So, to find ( angle B ), I subtract ( angle A ) from ( 180^circ ):[angle B = 180^circ - angle A = 180^circ - 55^circ = 125^circ]Thus, the measure of angle ( B ) is ( 125^circ ).

💡Okay, so I've got this problem here: I need to figure out how many functions exist from set A to set B, where A is {-1, 0} and B is {1, 2}. Hmm, functions... right, in math, a function assigns each element in the domain (which is set A here) to exactly one element in the codomain (which is set B). So, for each element in A, I have to choose an element in B to map it to.Let me start by listing out the elements in A and B. Set A has two elements: -1 and 0. Set B has two elements: 1 and 2. So, for each element in A, I have two choices in B. That seems straightforward.Wait, so for the first element in A, which is -1, I can map it to either 1 or 2. That's two options. Then, for the second element in A, which is 0, I can also map it to either 1 or 2. So, again, two options. Now, if I think about it, for each element in A, the number of choices in B is independent of the choices for the other elements. So, the total number of functions should be the product of the number of choices for each element. That means 2 choices for -1 and 2 choices for 0, so 2 times 2, which is 4.Let me double-check that. If I list out all possible functions, they would be:1. f(-1) = 1 and f(0) = 12. f(-1) = 1 and f(0) = 23. f(-1) = 2 and f(0) = 14. f(-1) = 2 and f(0) = 2Yep, that's four functions. Each function is uniquely determined by where it maps each element of A. Since there are two elements in A and two choices for each, the total number is 2^2 = 4.I think that makes sense. So, the number of functions from A to B is 4.

💡Okay, so I have this problem here: What is the value of (1296^{log_6 1728})^{1/4}? The options are A) 36, B) 216, C) 1296, D) 1728, and E) 6. Hmm, let me try to figure this out step by step.First, I know that 1296 and 1728 are both powers of 6, which might be helpful. Let me confirm that. I remember that 6^1 is 6, 6^2 is 36, 6^3 is 216, 6^4 is 1296, and 6^5 is 7776. So, yes, 1296 is 6^4, and 1728 is 6^3 because 6*6*6 is 216, and 216*6 is 1296, and 1296*6 is 7776. Wait, no, 6^3 is 216, so 1728 must be 12^3, right? Wait, no, 12^3 is 1728, but 6^3 is 216. Hmm, maybe I confused something.Wait, let me double-check. 6^1 is 6, 6^2 is 36, 6^3 is 216, 6^4 is 1296, 6^5 is 7776. So, 1728 is not a power of 6. Wait, but 1728 is 12^3, because 12*12 is 144, 144*12 is 1728. So, 1728 is 12^3, but 12 is 6*2, so maybe I can express 1728 in terms of 6 somehow. Let me see.Alternatively, maybe I can use logarithms to simplify this expression. The expression is (1296^{log_6 1728})^{1/4}. Let me break it down. First, let me compute log_6 1728. Since 1728 is 12^3, and 12 is 6*2, maybe I can write 1728 as (6*2)^3, which is 6^3 * 2^3, so 216*8=1728. So, 1728 is 6^3 * 2^3.But I need to find log_6 1728. Maybe I can use the change of base formula: log_6 1728 = frac{ln 1728}{ln 6}. But that might not be helpful here. Alternatively, since 1728 is 12^3, and 12 is 6*2, maybe I can express log_6 1728 as log_6 (12^3) = 3 log_6 12. And log_6 12 is log_6 (6*2) = log_6 6 + log_6 2 = 1 + log_6 2. So, log_6 1728 = 3(1 + log_6 2) = 3 + 3 log_6 2.Hmm, not sure if that helps. Maybe there's a better way. Let me think about the original expression: (1296^{log_6 1728})^{1/4}. Since 1296 is 6^4, I can rewrite 1296 as 6^4. So, the expression becomes ( (6^4)^{log_6 1728} )^{1/4}.Now, using the power of a power rule: (a^m)^n = a^{m*n}. So, (6^4)^{log_6 1728} = 6^{4 * log_6 1728}. Then, raising that to the 1/4 power, we get (6^{4 * log_6 1728})^{1/4} = 6^{(4 * log_6 1728) * (1/4)} = 6^{log_6 1728}.Wait, that simplifies nicely! Because 6^{log_6 1728} is just 1728, since a^{log_a b} = b. So, the entire expression simplifies to 1728^{1/4}? Wait, no, wait. Let me double-check.Wait, no, I think I made a mistake there. Let me go back. The expression is (1296^{log_6 1728})^{1/4}. I rewrote 1296 as 6^4, so it becomes (6^4)^{log_6 1728}. Then, using the power rule, that's 6^{4 * log_6 1728}. Then, raising that to the 1/4 power, it's (6^{4 * log_6 1728})^{1/4} = 6^{(4 * log_6 1728) * (1/4)} = 6^{log_6 1728}.Yes, that's correct. So, 6^{log_6 1728} is indeed 1728, because a^{log_a b} = b. So, the expression simplifies to 1728^{1/4}? Wait, no, wait. Wait, no, the exponent is already applied. Let me clarify.Wait, no, the expression is (1296^{log_6 1728})^{1/4} = 6^{log_6 1728} = 1728. But wait, that can't be right because 1728 is one of the options, option D. But let me check the steps again.Wait, no, I think I made a mistake in the exponentiation. Let me go through it again.Starting with (1296^{log_6 1728})^{1/4}.1296 is 6^4, so substitute that in: ( (6^4)^{log_6 1728} )^{1/4}.Using the power rule: (6^{4 * log_6 1728})^{1/4}.Now, when you raise a power to another power, you multiply the exponents: 6^{(4 * log_6 1728) * (1/4)}.Simplify the exponents: 4 * (1/4) = 1, so it becomes 6^{log_6 1728}.And as I said before, 6^{log_6 1728} = 1728.Wait, but 1728 is option D, but I have a feeling that the answer might be different. Let me check if I did everything correctly.Alternatively, maybe I can approach it differently. Let me compute log_6 1728 first. Since 1728 is 12^3, and 12 is 6*2, so 1728 = (6*2)^3 = 6^3 * 2^3 = 216 * 8 = 1728.So, log_6 1728 = log_6 (6^3 * 2^3) = log_6 6^3 + log_6 2^3 = 3 + 3 log_6 2.But I'm not sure if that helps. Alternatively, maybe I can express 1728 as 6^x and solve for x.So, 6^x = 1728.We know that 6^3 = 216, 6^4 = 1296, 6^5 = 7776. So, 1728 is between 6^3 and 6^4. Hmm, but that might not help directly.Wait, maybe I can use logarithms to find x.x = log_6 1728.But I think I'm going in circles here. Let me go back to the original expression.So, (1296^{log_6 1728})^{1/4}.I rewrote 1296 as 6^4, so it's (6^4)^{log_6 1728}.Then, using the power rule: 6^{4 * log_6 1728}.Then, raising that to the 1/4 power: 6^{(4 * log_6 1728) * (1/4)} = 6^{log_6 1728} = 1728.Wait, so the answer is 1728, which is option D. But I have a feeling that the answer might be different because 1728 is a large number, and the options include smaller numbers like 36, 216, etc.Wait, maybe I made a mistake in simplifying the exponents. Let me check again.Starting with (1296^{log_6 1728})^{1/4}.1296 = 6^4, so substitute: (6^4)^{log_6 1728}.Using the power rule: 6^{4 * log_6 1728}.Now, raising that to the 1/4 power: (6^{4 * log_6 1728})^{1/4} = 6^{(4 * log_6 1728) * (1/4)}.Simplify the exponent: 4 * (1/4) = 1, so it's 6^{log_6 1728}.And 6^{log_6 1728} = 1728.Wait, that seems correct. So, the answer should be 1728, which is option D.But let me double-check by calculating the numerical values.First, compute log_6 1728.We know that 6^3 = 216, 6^4 = 1296, 6^5 = 7776.1728 is between 6^3 and 6^4, closer to 6^4.Let me compute log_6 1728.We can write 1728 = 6^x.Taking natural logs: ln(1728) = x ln(6).So, x = ln(1728)/ln(6).Compute ln(1728): ln(1728) ≈ 7.453 (since e^7 ≈ 1096, e^7.4 ≈ 1700, so approx 7.45).ln(6) ≈ 1.7918.So, x ≈ 7.453 / 1.7918 ≈ 4.16.So, log_6 1728 ≈ 4.16.Now, compute 1296^{4.16}.But 1296 is 6^4, so 1296^{4.16} = (6^4)^{4.16} = 6^{16.64}.Then, take the 1/4 power: (6^{16.64})^{1/4} = 6^{4.16}.Wait, 6^{4.16} is approximately 6^4 * 6^0.16 ≈ 1296 * 1.12 ≈ 1450.But 1728 is 1728, which is larger than 1450. Hmm, that doesn't match. So, maybe my earlier conclusion was wrong.Wait, but I thought 6^{log_6 1728} = 1728, which is correct, but when I compute it numerically, I get a different result. That suggests I made a mistake in my reasoning.Wait, let me go back.Original expression: (1296^{log_6 1728})^{1/4}.Expressed as (6^4)^{log_6 1728} = 6^{4 * log_6 1728}.Then, (6^{4 * log_6 1728})^{1/4} = 6^{(4 * log_6 1728) * (1/4)} = 6^{log_6 1728}.But 6^{log_6 1728} is indeed 1728, so the expression simplifies to 1728^{1/4}? Wait, no, wait.Wait, no, the exponent is already applied. Let me clarify.Wait, no, the expression is (1296^{log_6 1728})^{1/4} = 6^{log_6 1728} = 1728.But when I computed numerically, I got approximately 1450, which is not 1728. So, that suggests an error in my reasoning.Wait, maybe I made a mistake in the exponent rules. Let me check.Starting with (a^b)^c = a^{b*c}.So, (1296^{log_6 1728})^{1/4} = 1296^{(log_6 1728) * (1/4)}.But 1296 is 6^4, so substitute: (6^4)^{(log_6 1728) * (1/4)} = 6^{4 * (log_6 1728) * (1/4)} = 6^{log_6 1728} = 1728.Wait, that's the same result as before. So, why does the numerical calculation give a different result?Wait, maybe I made a mistake in the numerical calculation. Let me try again.Compute log_6 1728.We know that 6^3 = 216, 6^4 = 1296, 6^5 = 7776.1728 is between 6^3 and 6^4.Let me compute log_6 1728.We can write 1728 = 6^x.Take natural logs: ln(1728) = x ln(6).Compute ln(1728): ln(1728) ≈ 7.453 (since e^7 ≈ 1096, e^7.4 ≈ 1700, so approx 7.45).ln(6) ≈ 1.7918.So, x ≈ 7.453 / 1.7918 ≈ 4.16.So, log_6 1728 ≈ 4.16.Now, compute 1296^{4.16}.But 1296 is 6^4, so 1296^{4.16} = (6^4)^{4.16} = 6^{16.64}.Then, take the 1/4 power: (6^{16.64})^{1/4} = 6^{4.16}.Now, 6^{4.16} is 6^4 * 6^0.16 ≈ 1296 * 1.12 ≈ 1450.But 1728 is 1728, which is larger than 1450. So, there's a discrepancy here.Wait, but according to the algebraic manipulation, the expression simplifies to 1728, but numerically, it's giving me approximately 1450. That suggests that my algebraic manipulation is wrong.Wait, let me check the algebra again.Original expression: (1296^{log_6 1728})^{1/4}.Express 1296 as 6^4: (6^4)^{log_6 1728}.Using the power rule: 6^{4 * log_6 1728}.Then, raise to the 1/4 power: (6^{4 * log_6 1728})^{1/4} = 6^{(4 * log_6 1728) * (1/4)} = 6^{log_6 1728}.And 6^{log_6 1728} = 1728.Wait, that seems correct. So, why is the numerical calculation giving me a different result?Wait, maybe I made a mistake in the numerical calculation. Let me try again.Compute log_6 1728.We know that 6^3 = 216, 6^4 = 1296, 6^5 = 7776.1728 is between 6^3 and 6^4.Let me compute log_6 1728.We can write 1728 = 6^x.Take natural logs: ln(1728) = x ln(6).Compute ln(1728): ln(1728) ≈ 7.453 (since e^7 ≈ 1096, e^7.4 ≈ 1700, so approx 7.45).ln(6) ≈ 1.7918.So, x ≈ 7.453 / 1.7918 ≈ 4.16.So, log_6 1728 ≈ 4.16.Now, compute 1296^{4.16}.But 1296 is 6^4, so 1296^{4.16} = (6^4)^{4.16} = 6^{16.64}.Then, take the 1/4 power: (6^{16.64})^{1/4} = 6^{4.16}.Now, 6^{4.16} is 6^4 * 6^0.16 ≈ 1296 * 1.12 ≈ 1450.Wait, but according to the algebraic manipulation, it should be 1728. So, where is the mistake?Wait, maybe I made a mistake in the order of operations. Let me check.Original expression: (1296^{log_6 1728})^{1/4}.Is it (1296^{log_6 1728})^{1/4} or 1296^{log_6 (1728^{1/4})}?Because exponentiation is right-associative, so it's (1296^{log_6 1728})^{1/4}.But if I interpret it as 1296^{log_6 (1728^{1/4})}, that would be different.Wait, let me clarify the order of operations.The expression is (1296^{log_6 1728})^{1/4}.So, first compute 1296^{log_6 1728}, then take the 1/4 power.Alternatively, if it were 1296^{log_6 (1728^{1/4})}, that would be different.But in the given expression, it's (1296^{log_6 1728})^{1/4}.So, the exponent is log_6 1728, and then the entire result is raised to the 1/4 power.So, my initial algebraic manipulation is correct.But then why does the numerical calculation give a different result?Wait, maybe I made a mistake in the numerical calculation.Let me compute 6^{log_6 1728}.That should be 1728, right?Yes, because a^{log_a b} = b.So, 6^{log_6 1728} = 1728.Therefore, the expression simplifies to 1728.But when I computed it numerically, I got approximately 1450, which is not 1728.Wait, that suggests that my numerical calculation is wrong.Wait, let me compute 6^{log_6 1728} numerically.Compute log_6 1728 ≈ 4.16.Then, 6^{4.16} ≈ 6^4 * 6^0.16 ≈ 1296 * 1.12 ≈ 1450.But 6^{log_6 1728} should be 1728.Wait, that's a contradiction.Wait, no, actually, 6^{log_6 1728} is indeed 1728, so my numerical calculation must be wrong.Wait, let me compute 6^4.16 more accurately.Compute 6^4 = 1296.Compute 6^0.16.We can write 6^0.16 = e^{0.16 * ln 6} ≈ e^{0.16 * 1.7918} ≈ e^{0.2867} ≈ 1.332.So, 6^4.16 ≈ 1296 * 1.332 ≈ 1728.Ah, there we go. I must have miscalculated earlier.So, 6^0.16 ≈ 1.332, not 1.12.Therefore, 6^4.16 ≈ 1296 * 1.332 ≈ 1728.So, that matches the algebraic result.Therefore, the expression simplifies to 1728.But wait, 1728 is one of the options, option D.But earlier, I thought the answer might be different because I made a mistake in the numerical calculation.So, the correct answer is 1728, which is option D.Wait, but let me check again.Alternatively, maybe I can approach it differently.Let me compute log_6 1728.We know that 1728 = 12^3, and 12 = 6 * 2.So, log_6 1728 = log_6 (12^3) = 3 log_6 12.And log_6 12 = log_6 (6 * 2) = log_6 6 + log_6 2 = 1 + log_6 2.So, log_6 1728 = 3(1 + log_6 2) = 3 + 3 log_6 2.Now, let's compute 1296^{log_6 1728}.1296 is 6^4, so 1296^{log_6 1728} = (6^4)^{log_6 1728} = 6^{4 * log_6 1728}.Substitute log_6 1728 = 3 + 3 log_6 2.So, 6^{4 * (3 + 3 log_6 2)} = 6^{12 + 12 log_6 2}.Now, 6^{12 + 12 log_6 2} = 6^{12} * 6^{12 log_6 2}.But 6^{12 log_6 2} = (6^{log_6 2})^{12} = 2^{12} = 4096.So, 6^{12} * 4096.But 6^{12} is (6^6)^2 = (46656)^2 = 2176782336.Wait, that's a huge number. Then, 2176782336 * 4096 is even larger.But that can't be right because earlier, we saw that 6^{log_6 1728} = 1728, so the expression should simplify to 1728.Wait, I think I made a mistake in this approach.Wait, let me go back.We have 6^{4 * log_6 1728}.But 4 * log_6 1728 = log_6 1728^4.So, 6^{log_6 1728^4} = 1728^4.Wait, but 1728^4 is a huge number, and then taking the 1/4 power would give 1728.Wait, that makes sense.So, (1296^{log_6 1728})^{1/4} = (6^{log_6 1728^4})^{1/4} = (1728^4)^{1/4} = 1728^{(4 * 1/4)} = 1728^1 = 1728.Yes, that's correct.So, the answer is 1728, which is option D.But wait, I'm confused because earlier, when I tried to compute it numerically, I initially got a wrong result, but then corrected myself.So, to summarize, the expression simplifies to 1728, which is option D.

💡First, I recognize that the tangent of π minus an angle is equal to the negative tangent of that angle. So, if tan(π - α) equals 2, then tan(α) must be -2.Next, I recall the double angle formula for tangent, which is tan(2α) equals (2 tan α) divided by (1 minus tan squared α). Substituting tan α with -2 in the formula, I calculate the numerator as 2 times -2, which is -4. For the denominator, I calculate 1 minus (-2) squared, which is 1 minus 4, resulting in -3.Finally, dividing the numerator by the denominator gives me -4 divided by -3, which simplifies to 4/3.

💡Okay, so I have this function f(x) = (-4x + 1)e^x, and I need to find its maximum value on the interval [0, 1]. Hmm, I remember from my calculus class that to find maximum or minimum values of a function on a closed interval, I should check the critical points inside the interval and also evaluate the function at the endpoints. First, I need to find the critical points by taking the derivative of f(x) and setting it equal to zero. Let me recall how to differentiate functions like this. Since f(x) is a product of two functions, (-4x + 1) and e^x, I should use the product rule. The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is u'(x)v(x) + u(x)v'(x).So, let me set u(x) = (-4x + 1) and v(x) = e^x. Then, u'(x) would be the derivative of (-4x + 1), which is -4, and v'(x) is the derivative of e^x, which is still e^x. Applying the product rule, f'(x) = u'(x)v(x) + u(x)v'(x) = (-4)e^x + (-4x + 1)e^x. Wait, let me write that out more clearly: f'(x) = (-4)e^x + (-4x + 1)e^x. Now, I can factor out e^x from both terms since it's common. So, f'(x) = e^x*(-4 + (-4x + 1)). Simplifying inside the parentheses: -4 + (-4x + 1) = -4 - 4x + 1 = (-4 + 1) - 4x = -3 - 4x. Therefore, f'(x) = e^x*(-4x - 3). Since e^x is always positive for any real x, the sign of f'(x) depends on the other factor, which is (-4x - 3). Let me analyze this. If I set f'(x) = 0, then e^x*(-4x - 3) = 0. But e^x is never zero, so the only solution comes from -4x - 3 = 0. Solving for x: -4x - 3 = 0 => -4x = 3 => x = -3/4. Hmm, x = -3/4 is a critical point, but it's not in the interval [0, 1]. That means there are no critical points within the interval where the function could have a maximum or minimum. Therefore, the maximum value must occur at one of the endpoints of the interval, which are x = 0 and x = 1. Let me evaluate f(x) at both endpoints. First, at x = 0: f(0) = (-4*0 + 1)e^0 = (0 + 1)*1 = 1. Next, at x = 1: f(1) = (-4*1 + 1)e^1 = (-4 + 1)e = (-3)e ≈ -3*2.718 ≈ -8.154. Comparing these two values, f(0) = 1 and f(1) ≈ -8.154. Clearly, 1 is the larger value. Therefore, the maximum value of the function f(x) on the interval [0, 1] is 1, occurring at x = 0. Wait, just to make sure I didn't make any mistakes, let me double-check my calculations. First, the derivative: f'(x) = e^x*(-4x - 3). Correct, because the product rule was applied properly. Setting f'(x) = 0: e^x*(-4x - 3) = 0. Since e^x isn't zero, solving -4x - 3 = 0 gives x = -3/4. That's outside our interval, so no critical points inside [0,1]. Evaluating f(0): (-4*0 + 1)e^0 = 1*1 = 1. Correct. Evaluating f(1): (-4*1 + 1)e^1 = (-3)e ≈ -8.154. Correct. So, yes, the maximum is indeed at x = 0 with a value of 1. I think that's thorough enough. I don't see any errors in my reasoning or calculations.