💡Okay, so I have this problem where I need to figure out how many integers k between 2 and 2020 inclusive satisfy the condition that k+1 divides a_k - 1. The sequence {a_n} is defined recursively with a_1 = 1, a_2 = 2, and for n ≥ 2, a_{n+1} = (n+1)^{a_n}. Hmm, that seems pretty intense because each term is an exponentiation of the next number raised to the previous term. So, the sequence grows extremely rapidly.First, let me try to understand the sequence a bit better. Starting with a_1 = 1, then a_2 = 2. Then a_3 = 3^{a_2} = 3^2 = 9. Next, a_4 = 4^{a_3} = 4^9, which is 262144. Wow, that's already huge. a_5 would be 5^{262144}, which is an astronomically large number. So, for k up to 2020, a_k is going to be unimaginably large. But maybe I don't need to compute a_k directly; perhaps there's a pattern or some modular arithmetic trick I can use.The problem is asking about divisibility: k+1 divides a_k - 1. In other words, a_k ≡ 1 mod (k+1). So, I need to find all k between 2 and 2020 such that a_k ≡ 1 mod (k+1). Given that a_{n+1} = (n+1)^{a_n}, let's see if I can find a pattern or some property that can help me evaluate a_k modulo (k+1). Maybe Euler's theorem or Fermat's little theorem could be useful here because they deal with exponents modulo some number.But before jumping into theorems, let me consider the parity of a_k. Since a_1 is 1 (odd), a_2 is 2 (even), a_3 is 3^2 = 9 (odd), a_4 is 4^9, which is even because 4 is even and any power of an even number is even. Similarly, a_5 is 5^{a_4}, which is odd because 5 is odd and any power of an odd number is odd. So, it seems like a_n is odd if n is odd and even if n is even. That's an interesting pattern.So, a_k is odd when k is odd and even when k is even. Now, considering the condition a_k ≡ 1 mod (k+1). If k is even, then k+1 is odd. If k is odd, then k+1 is even. Let's see:Case 1: k is even. Then k+1 is odd. Since a_k is even (because k is even), we have an even number minus 1, which is odd. So, a_k - 1 is odd. But k+1 is odd, so it's possible for an odd number to divide another odd number. But I don't know if that's necessarily true.Case 2: k is odd. Then k+1 is even. Since a_k is odd (because k is odd), a_k - 1 is even. So, a_k - 1 is divisible by 2, but k+1 is even, so maybe it's possible for k+1 to divide a_k - 1.But wait, let's think about the parity. If k is even, a_k is even, so a_k - 1 is odd. Since k+1 is odd, it's possible for k+1 to divide an odd number. If k is odd, a_k is odd, so a_k - 1 is even, and k+1 is even, so it's possible for k+1 to divide an even number. So, both cases are possible, but maybe one case is more likely than the other.But maybe there's a deeper pattern here. Let me think about the recursive definition. Since a_{n+1} = (n+1)^{a_n}, perhaps I can use induction or some modular properties.Let me try small values of k to see if I can spot a pattern.For k=2: a_2 = 2. We need to check if 3 divides 2 - 1 = 1. 3 does not divide 1, so k=2 doesn't work.For k=3: a_3 = 9. Check if 4 divides 9 - 1 = 8. Yes, 4 divides 8, so k=3 works.For k=4: a_4 = 4^9 = 262144. Check if 5 divides 262144 - 1 = 262143. Let's see, 262143 divided by 5 is 52428.6, which is not an integer, so 5 does not divide 262143. So, k=4 doesn't work.For k=5: a_5 = 5^{262144}. That's a huge number, but maybe I can compute it modulo 6. Since 5 ≡ -1 mod 6, so 5^{262144} ≡ (-1)^{262144} ≡ 1 mod 6. So, a_5 ≡ 1 mod 6, which means 6 divides a_5 - 1. So, k=5 works.Wait, that's interesting. For k=3 and k=5, which are odd, the condition holds. For k=2 and k=4, which are even, it doesn't hold. Maybe this is a pattern.Let me test k=6: a_6 = 6^{a_5}. Since a_5 is huge, but let's compute a_6 modulo 7. 6 ≡ -1 mod 7, so 6^{a_5} ≡ (-1)^{a_5} mod 7. Now, a_5 is 5^{262144}, which is an odd exponent because 5 is odd and any power of an odd number is odd. So, (-1)^{odd} = -1 mod 7. Therefore, a_6 ≡ -1 mod 7, which means a_6 - 1 ≡ -2 mod 7. So, 7 does not divide a_6 - 1. Therefore, k=6 doesn't work.Similarly, for k=7: a_7 = 7^{a_6}. Let's compute a_7 modulo 8. 7 ≡ -1 mod 8, so 7^{a_6} ≡ (-1)^{a_6} mod 8. Now, a_6 is 6^{a_5}, which is even because 6 is even and any power of an even number is even. So, (-1)^{even} = 1 mod 8. Therefore, a_7 ≡ 1 mod 8, which means 8 divides a_7 - 1. So, k=7 works.Hmm, so it seems like when k is odd, the condition holds, and when k is even, it doesn't. Let me check k=1 just to see, even though the problem starts at k=2. a_1=1, so 2 divides 1-1=0, which is true, but the problem starts at k=2.So, if this pattern holds, then all odd k between 2 and 2020 inclusive satisfy the condition. Let's see how many odd numbers are there between 2 and 2020 inclusive.The smallest odd number ≥2 is 3, and the largest odd number ≤2020 is 2019. So, the sequence is 3,5,7,...,2019. This is an arithmetic sequence with first term 3, last term 2019, common difference 2.The number of terms is given by ((last term - first term)/common difference) + 1 = ((2019 - 3)/2) + 1 = (2016/2) + 1 = 1008 + 1 = 1009.Therefore, there are 1009 integers k between 2 and 2020 inclusive such that k+1 divides a_k -1.But wait, let me make sure this pattern holds for all k. I tested k=2,3,4,5,6,7 and it worked for k=3,5,7. Let me try k=8 and k=9 to see.For k=8: a_8 = 8^{a_7}. Compute a_8 modulo 9. 8 ≡ -1 mod 9, so 8^{a_7} ≡ (-1)^{a_7} mod 9. Now, a_7 is 7^{a_6}, which is odd because 7 is odd and any power of an odd number is odd. So, (-1)^{odd} = -1 mod 9. Therefore, a_8 ≡ -1 mod 9, so a_8 -1 ≡ -2 mod 9. 9 does not divide -2, so k=8 doesn't work.For k=9: a_9 = 9^{a_8}. Compute a_9 modulo 10. 9 ≡ -1 mod 10, so 9^{a_8} ≡ (-1)^{a_8} mod 10. Now, a_8 is 8^{a_7}, which is even because 8 is even and any power of an even number is even. So, (-1)^{even} = 1 mod 10. Therefore, a_9 ≡ 1 mod 10, so 10 divides a_9 -1. Hence, k=9 works.So, the pattern continues. It seems that whenever k is odd, the condition holds, and when k is even, it doesn't. Therefore, the number of such k is equal to the number of odd integers between 2 and 2020 inclusive, which is 1009.

💡First, I need to evaluate the expression (frac{10^{-3} cdot 2^2}{10^{-4}}).I'll start by simplifying the exponents. (10^{-3}) is equal to (frac{1}{10^3}), and (10^{-4}) is equal to (frac{1}{10^4}).Next, I'll calculate (2^2), which is 4.Now, the expression becomes (frac{frac{4}{10^3}}{frac{1}{10^4}}).To divide these fractions, I'll multiply the numerator by the reciprocal of the denominator: (frac{4}{10^3} times 10^4).This simplifies to (4 times frac{10^4}{10^3}).Since (10^4) divided by (10^3) is (10^{4-3} = 10^1 = 10), the expression becomes (4 times 10 = 40).Therefore, the final answer is 40.

💡First, I need to understand the given ratio of products A, B, and C, which is 5:2:3. This means that for every 5 units of product A produced, there are 2 units of product B and 3 units of product C.Next, since stratified sampling is being used, the proportion of each product in the sample should reflect the same ratio as in the overall production. This means that the proportion of Type A products in the sample should be the same as their proportion in the entire production.To find the total number of parts in the ratio, I add up the parts: 5 + 2 + 3 = 10. This tells me that product A makes up 5 out of every 10 products.Given that there are 15 Type A products in the sample, I can set up a proportion to find the total sample size ( n ). The proportion of Type A products in the sample should be equal to the proportion in the overall production:[frac{15}{n} = frac{5}{10}]Simplifying the ratio on the right side:[frac{15}{n} = frac{1}{2}]To solve for ( n ), I can cross-multiply:[15 times 2 = n][n = 30]Therefore, the total sample size ( n ) is 30.

💡Okay, so I have this system of inequalities about x:1. x + 5 > 02. x - m ≤ 1And I need to find the range of the real number m such that this system has exactly 3 integer solutions. Hmm, okay, let me try to break this down step by step.First, let's solve each inequality separately to understand the range of x.Starting with the first inequality: x + 5 > 0. If I subtract 5 from both sides, I get x > -5. So, x has to be greater than -5.Now, the second inequality: x - m ≤ 1. If I add m to both sides, I get x ≤ m + 1. So, x has to be less than or equal to m + 1.Putting these two together, the solution for x is: -5 < x ≤ m + 1.So, x is greater than -5 and less than or equal to m + 1. Now, we need to find the integer solutions for x in this interval. The problem states that there are exactly 3 integer solutions. So, I need to figure out what integers lie between -5 and m + 1, and ensure that there are exactly 3 of them.Let me list the integers greater than -5. They are: -4, -3, -2, -1, 0, 1, 2, and so on. But since x has to be less than or equal to m + 1, the upper bound is m + 1. So, depending on the value of m, the upper bound will change, which affects how many integers are in the interval.We need exactly 3 integer solutions. Let's think about what integers these could be. Since x has to be greater than -5, the smallest integer x can be is -4. So, if we have exactly 3 integers, they must be -4, -3, and -2. That's three integers. So, the next integer after -2 is -1, which should not be included because we only want 3 solutions.Therefore, the upper bound m + 1 must be such that it includes -2 but excludes -1. So, m + 1 must be greater than or equal to -2 but less than -1. Wait, let me think about that again.If m + 1 is greater than or equal to -2, then x can be -2. But if m + 1 is less than -1, then x cannot be -1. So, combining these two, m + 1 must be in the interval [-2, -1). That is, m + 1 is greater than or equal to -2 and less than -1.So, if I solve for m, subtract 1 from all parts of the inequality:-2 ≤ m + 1 < -1Subtracting 1:-3 ≤ m < -2So, m must be greater than or equal to -3 and less than -2.Let me verify this. If m is -3, then m + 1 is -2. So, the upper bound is -2. So, x can be -4, -3, -2. That's 3 integers. If m is just less than -2, say m = -2.5, then m + 1 is -1.5. So, x can be -4, -3, -2, because -1.5 is less than -1, so x cannot be -1. So, again, 3 integers. If m is exactly -2, then m + 1 is -1, so x can be up to -1, which would include -4, -3, -2, -1, which is 4 integers, which is too many. So, m has to be less than -2.Similarly, if m is less than -3, say m = -4, then m + 1 is -3. So, x can be -4, -3, but not -2 because -3 is less than -2. Wait, that's only 2 integers, which is too few. So, m can't be less than -3.Therefore, m must be between -3 and -2, including -3 but not including -2. So, the range of m is -3 ≤ m < -2.I think that makes sense. Let me just recap:1. Solve each inequality to find the range of x: -5 < x ≤ m + 1.2. Identify the integer solutions within this range.3. Determine that exactly 3 integers (-4, -3, -2) must be included.4. Set the upper bound m + 1 such that it includes -2 but excludes -1.5. Solve for m to find the range: -3 ≤ m < -2.Yes, that seems correct. I don't think I missed anything. So, the range of m is from -3 to -2, including -3 but not including -2.**Final Answer**The range of real number ( m ) is boxed{[-3, -2)}.

💡Okay, so I have this problem about a trapezoid PQRS. Let me try to visualize it first. A trapezoid has one pair of sides that are parallel. Here, PQ and RS are the parallel sides because it's a trapezoid. They give me the lengths of PQ and RS, which are 40 and 25 respectively. The altitude from PQ to RS is 10, which means the height of the trapezoid is 10 units. Also, they mention that PR is 20. I need to find the area of this trapezoid.Hmm, the area of a trapezoid is usually calculated using the formula: (1/2) * (sum of the two parallel sides) * height. So, if I can confirm the height and the lengths of the two bases, I can plug them into the formula. But wait, they also gave me PR = 20. I wonder if that's necessary for finding the area or if it's just extra information. Maybe it's needed to confirm something else about the trapezoid.Let me draw a rough sketch of the trapezoid. Let's assume PQ is the top base and RS is the bottom base. So, PQ is 40 units long, RS is 25 units, and the height is 10 units. The sides are PS and QR. PR is one of the diagonals, connecting P to R, and it's 20 units long.I think I need to figure out if the trapezoid is isosceles or not. If it's isosceles, then the non-parallel sides are equal, but I don't think that's necessarily the case here. Since they gave me the length of the diagonal, maybe I can use that to find some missing information.Let me think about the right triangles formed by the height. If I drop a perpendicular from P to RS, let's call the foot of this perpendicular T. Similarly, if I drop a perpendicular from Q to RS, let's call that foot U. So, PT and QU are both 10 units, the height of the trapezoid.Now, the length of the top base PQ is 40, and the bottom base RS is 25. The difference in the lengths of the two bases is 15 units. This means that the total extra length on the top base compared to the bottom base is 15 units. Since the trapezoid is a quadrilateral, this extra length is distributed equally on both sides if it's isosceles, but since we don't know that, maybe we can use the diagonal PR to find out.Wait, PR is 20 units. So, in triangle PRT, we have PR = 20, PT = 10, and RT is part of the base RS. Let me denote RT as x. Then, in triangle PRT, by the Pythagorean theorem, we have:PR² = PT² + RT²20² = 10² + x²400 = 100 + x²x² = 300x = sqrt(300) = 10*sqrt(3)So, RT is 10*sqrt(3). Similarly, if I consider the other side, the length from S to U would be something else. Wait, maybe I'm complicating things.Alternatively, since the difference in the bases is 15, and if I assume that the extra length is split equally on both sides, then each side would have an extension of 7.5 units. But since we have the diagonal PR, which is 20, maybe we can use that to find the actual extensions.Wait, let's think about the coordinates. Maybe assigning coordinates to the points can help. Let me place point P at (0, 0). Then, since PQ is 40 units, point Q would be at (40, 0). The height is 10, so the base RS is 10 units below. Let me denote point R as (a, 10) and point S as (b, 10). Since RS is 25 units, the distance between R and S is 25, so |b - a| = 25.Now, the diagonal PR connects P(0,0) to R(a,10) and has a length of 20. So, using the distance formula:PR = sqrt[(a - 0)² + (10 - 0)²] = sqrt(a² + 100) = 20So, sqrt(a² + 100) = 20Squaring both sides: a² + 100 = 400a² = 300a = sqrt(300) = 10*sqrt(3) or a = -10*sqrt(3)Since we're dealing with a trapezoid, and assuming it's oriented so that the bases are horizontal, a should be positive. So, a = 10*sqrt(3). Therefore, point R is at (10*sqrt(3), 10).Now, since RS is 25 units long, and R is at (10*sqrt(3), 10), then point S must be at (10*sqrt(3) + 25, 10). Wait, but hold on, because the top base PQ is 40 units, and the bottom base RS is 25 units. So, the projection of PQ onto RS is shorter by 15 units. This means that the sides PS and QR are slanting inwards.Wait, maybe I should calculate the coordinates of S. If R is at (10*sqrt(3), 10), then S is at (10*sqrt(3) + 25, 10). But then, the length from Q(40,0) to S(10*sqrt(3) + 25,10) should be equal to the length of QR. Hmm, but I don't know QR's length.Alternatively, maybe I can find the coordinates of S in another way. Since the top base is 40 and the bottom base is 25, the difference is 15. So, if I project PQ down to RS, the overhang on each side would be (40 - 25)/2 = 7.5 units on each side. But wait, that's only if the trapezoid is isosceles. Since we don't know that, maybe the overhangs are different.But wait, we have the diagonal PR, which is 20. So, maybe the overhang on the left side is 10*sqrt(3), as we found earlier, and the overhang on the right side would be 40 - 25 - 10*sqrt(3). Wait, that might not make sense.Let me think again. If I have point P at (0,0), Q at (40,0), R at (10*sqrt(3),10), and S at (10*sqrt(3) + 25,10). Then, the side PS connects P(0,0) to S(10*sqrt(3) + 25,10). Let me calculate the length of PS.PS = sqrt[(10*sqrt(3) + 25 - 0)² + (10 - 0)²]= sqrt[(10*sqrt(3) + 25)² + 100]This seems complicated, but maybe it's not necessary. Since the problem is asking for the area, and I already know the two bases and the height, maybe I can just use the area formula without worrying about the sides.Wait, the area formula for a trapezoid is (1/2)*(PQ + RS)*height. So, plugging in the values:Area = (1/2)*(40 + 25)*10 = (1/2)*65*10 = 325.But wait, the answer choices include 325 as option B. So, is it that straightforward? Did I need to use the diagonal PR at all? It seems like I didn't need it because the area formula only requires the two bases and the height, which were given. But why did they give me PR = 20? Maybe to confirm that the trapezoid is possible with those dimensions or to ensure that the height is indeed 10.Wait, maybe I made a mistake in assuming the height is 10. Let me double-check. The altitude from PQ to RS is 10, so that should be the height. So, the area should indeed be (1/2)*(40 + 25)*10 = 325.But just to be thorough, let me see if the diagonal PR being 20 affects the height. If PR is 20, and the height is 10, then in triangle PRT, where T is the foot of the perpendicular from P to RS, we have PT = 10 and PR = 20. So, RT, the horizontal component, would be sqrt(20² - 10²) = sqrt(400 - 100) = sqrt(300) = 10*sqrt(3). So, RT is 10*sqrt(3), which is approximately 17.32.But the total length of RS is 25, so the remaining part from T to S would be 25 - 10*sqrt(3). That seems fine. So, the overhang on the left side is 10*sqrt(3), and on the right side, it's 25 - 10*sqrt(3). But since 10*sqrt(3) is about 17.32, which is more than half of 25, that might mean that the trapezoid is not symmetric, but that's okay.Wait, but if the overhang on the left is 10*sqrt(3) and on the right is 25 - 10*sqrt(3), then the total overhang is 10*sqrt(3) + (25 - 10*sqrt(3)) = 25, which is less than the top base of 40. Wait, that doesn't make sense because the top base is longer than the bottom base by 15 units. So, the overhangs should add up to 15 units.Wait, maybe I'm confusing something. Let me clarify. The top base PQ is 40, and the bottom base RS is 25. So, the difference is 15 units. This means that the total overhang on both sides combined is 15 units. So, if the overhang on the left is x, then the overhang on the right is 15 - x.But from the diagonal PR, we found that RT is 10*sqrt(3), which is approximately 17.32. But that's more than 15, which contradicts the overhang. Hmm, that doesn't make sense. Maybe I made a mistake in my earlier assumption.Wait, no, RT is the horizontal distance from R to T, which is the foot of the perpendicular from P. So, if R is at (10*sqrt(3),10), then T is at (10*sqrt(3),0). But PQ is from (0,0) to (40,0). So, the overhang on the left side is 10*sqrt(3), and the overhang on the right side is 40 - (10*sqrt(3) + 25). Wait, 40 - (10*sqrt(3) + 25) = 15 - 10*sqrt(3). But 10*sqrt(3) is about 17.32, so 15 - 17.32 is negative, which doesn't make sense. That means the overhang on the right side is negative, which implies that the bottom base RS extends beyond the top base PQ on the right side.Wait, that can't be because RS is shorter than PQ. So, maybe my coordinate system is flipped. Maybe R is on the left side and S is on the right side. Let me try that.Let me place point P at (0,0), Q at (40,0), R at (a,10), and S at (b,10). Since RS is 25, b - a = 25. The diagonal PR is 20, so from P(0,0) to R(a,10):PR = sqrt(a² + 10²) = 20So, a² + 100 = 400a² = 300a = sqrt(300) = 10*sqrt(3) ≈ 17.32So, R is at (10*sqrt(3),10). Then, S is at (10*sqrt(3) + 25,10). Now, the overhang on the left side is 10*sqrt(3) ≈ 17.32, and the overhang on the right side is 40 - (10*sqrt(3) + 25) = 15 - 10*sqrt(3) ≈ 15 - 17.32 = -2.32. Negative overhang means that the bottom base extends beyond the top base on the right side by 2.32 units. But since RS is shorter than PQ, this shouldn't happen. So, maybe my initial assumption about the placement is wrong.Alternatively, maybe the trapezoid is not convex, but that's unlikely. Wait, perhaps I should consider that the diagonal PR is not the one from P to R, but from P to S? Wait, no, PR is from P to R.Wait, maybe I need to adjust the coordinate system. Let me try placing R at (x,10) and S at (x + 25,10). Then, the diagonal PR connects P(0,0) to R(x,10), which is 20 units. So:sqrt(x² + 10²) = 20x² + 100 = 400x² = 300x = sqrt(300) = 10*sqrt(3) ≈ 17.32So, R is at (10*sqrt(3),10), and S is at (10*sqrt(3) + 25,10). Now, the top base PQ is from (0,0) to (40,0). So, the overhang on the left is 10*sqrt(3) ≈ 17.32, and the overhang on the right is 40 - (10*sqrt(3) + 25) ≈ 40 - 42.32 ≈ -2.32. Again, negative overhang on the right.This suggests that the bottom base RS extends beyond the top base PQ on the right side by about 2.32 units. But since RS is shorter than PQ, this seems contradictory. Wait, no, actually, RS is shorter, so the overhang should be on both sides, but the total overhang is 15 units. But in this case, the overhang on the left is 17.32, which is more than the total overhang of 15. That doesn't make sense.Wait, maybe I'm misunderstanding the overhang. The overhang is the amount by which the top base extends beyond the bottom base on each side. So, if the top base is longer by 15 units, the overhang on each side should add up to 15. But in this case, the overhang on the left is 10*sqrt(3) ≈ 17.32, which is more than 15, which is impossible. Therefore, there must be a mistake in my approach.Wait, perhaps the diagonal PR is not from P to R, but from P to S? Let me check the problem statement again. It says PR = 20. So, P to R is 20. So, my initial approach was correct.Wait, maybe the height is not 10. But the problem says the altitude from PQ to RS is 10. So, the height is indeed 10. Hmm, this is confusing.Wait, let me try a different approach. The area of the trapezoid is (1/2)*(PQ + RS)*height = (1/2)*(40 + 25)*10 = 325. So, regardless of the diagonal, the area is 325. But why did they give me PR = 20? Maybe to confirm that such a trapezoid exists with those dimensions.Alternatively, maybe the height is not 10, but the altitude is 10, which is the same as the height. So, the area should be 325. Therefore, the answer is 325, which is option B.I think I'm overcomplicating this. The area formula only requires the two bases and the height, which are given. The diagonal PR is probably there to ensure that the trapezoid is possible with those dimensions, but it doesn't affect the area calculation. So, I'll go with 325.

💡Okay, so I have this function f(x) = a^x + m b^x, and there are three parts to the problem. Let me take them one by one.Starting with part (1): a = 2, b = 1/2, and f(x) is an even function on R. I need to find m.Hmm, an even function satisfies f(-x) = f(x) for all x. So, I can write that 2^{-x} + m (1/2)^{-x} = 2^x + m (1/2)^x.Wait, (1/2)^{-x} is the same as 2^x, right? Because (1/2)^{-x} = (2^{-1})^{-x} = 2^{x}.So, substituting that in, the equation becomes 2^{-x} + m * 2^x = 2^x + m * 2^{-x}.Let me rearrange terms: 2^{-x} - m * 2^{-x} = 2^x - m * 2^x.Factor out 2^{-x} on the left and 2^x on the right: (1 - m) 2^{-x} = (1 - m) 2^x.Hmm, so either 1 - m = 0, which would mean m = 1, or 2^{-x} = 2^x for all x, which is not possible unless x = 0. But since f(x) is even for all x, the only possibility is that 1 - m = 0, so m = 1.Wait, let me check that again. If I plug m = 1 into f(x), I get f(x) = 2^x + (1)(1/2)^x = 2^x + 2^{-x}, which is indeed even because f(-x) = 2^{-x} + 2^{x} = f(x). So that makes sense.Okay, so part (1) seems to be m = 1.Moving on to part (2): a = 4, b = 2, and f(x) has a minimum value on R. I need to find the range of m.So f(x) = 4^x + m 2^x. Let me rewrite this in terms of t = 2^x, since 4^x = (2^2)^x = 2^{2x} = t^2. So f(x) becomes t^2 + m t, where t > 0 because 2^x is always positive.Now, f(t) = t^2 + m t. This is a quadratic function in t. For it to have a minimum, the quadratic must open upwards, which it does since the coefficient of t^2 is positive. However, the vertex of the parabola is at t = -m/(2*1) = -m/2.But since t > 0, the vertex must lie within the domain t > 0 for the function to have a minimum. So, -m/2 > 0, which implies that m < 0.Therefore, the range of m is all real numbers less than 0.Wait, let me think again. If m is negative, then the vertex is at t = -m/2, which is positive because m is negative. So, yes, the function f(t) will have a minimum at that t. If m were positive, the vertex would be at a negative t, which is not in our domain, so the function would be increasing for t > 0, hence no minimum. So, yes, m must be less than 0.So part (2) is m < 0.Now, part (3): For a ∈ (0,1) and b > 1, solve the inequality f(x) > 0 in terms of x.So f(x) = a^x + m b^x > 0.Given that a is between 0 and 1, a^x is a decreasing function, and since b > 1, b^x is an increasing function.I need to find the values of x for which a^x + m b^x > 0.Let me rewrite the inequality: a^x > -m b^x.Divide both sides by b^x (which is positive, so inequality sign doesn't change): (a/b)^x > -m.Now, since a ∈ (0,1) and b > 1, a/b is less than 1, so (a/b)^x is a decreasing function.So, (a/b)^x > -m.Now, this depends on the value of m.Case 1: If m ≥ 0, then -m ≤ 0. Since (a/b)^x is always positive (because a and b are positive), the inequality (a/b)^x > -m is always true. So, for m ≥ 0, the inequality holds for all x.Case 2: If m < 0, then -m > 0. So, we have (a/b)^x > -m.Since (a/b)^x is decreasing, the inequality (a/b)^x > -m is equivalent to x < log_{a/b}(-m).But wait, since a/b < 1, the logarithm base a/b is a decreasing function. So, to solve (a/b)^x > -m, we take logarithms on both sides, remembering that log_{a/b} is decreasing, so the inequality flips.So, x < log_{a/b}(-m).But log_{a/b}(-m) can be written as ln(-m)/ln(a/b). Since a/b < 1, ln(a/b) is negative, so log_{a/b}(-m) is positive if -m > 1, and negative otherwise.Wait, but regardless, the solution is x < log_{a/b}(-m).So, summarizing:- If m ≥ 0, f(x) > 0 for all x ∈ R.- If m < 0, f(x) > 0 for x < log_{a/b}(-m).Let me check with an example. Suppose a = 1/2, b = 2, m = -1.Then f(x) = (1/2)^x - 2^x.We want (1/2)^x - 2^x > 0.Which is equivalent to (1/2)^x > 2^x.Which is equivalent to (1/2)^x / 2^x > 1, which is (1/4)^x > 1.Which is equivalent to x < 0, because (1/4)^x is decreasing.And log_{a/b}(-m) = log_{(1/2)/2}(-(-1)) = log_{1/4}(1) = 0, since (1/4)^0 = 1.So, x < 0, which matches our expectation.Another example: a = 1/3, b = 3, m = -2.Then f(x) = (1/3)^x - 2*3^x.We want (1/3)^x - 2*3^x > 0.Which is (1/3)^x > 2*3^x.Divide both sides by 3^x: (1/3)^x / 3^x = (1/3)^{2x} = (1/9)^x > 2.So, (1/9)^x > 2.Take natural logs: x ln(1/9) > ln 2.Since ln(1/9) is negative, divide both sides and reverse inequality: x < ln 2 / ln(1/9) = ln 2 / (-ln 9) = - ln 2 / ln 9.Which is the same as log_{1/9}(2) = log_{a/b}(-m) because a/b = 1/3 / 3 = 1/9, and -m = 2.So, x < log_{1/9}(2), which is consistent with our solution.Therefore, the solution seems correct.So, putting it all together:1. m = 12. m < 03. If m ≥ 0, all real x; if m < 0, x < log_{a/b}(-m)