answer：Okay, so I have this problem here where I need to find f(2) for an odd function f(x). The function is defined on all real numbers, and they've given me the expression for f(x) when x is in the interval (-∞, 0). Specifically, for x < 0, f(x) = 2x³ + x². Hmm, okay.First, I remember that an odd function has a specific property: f(-x) = -f(x) for all x in its domain. That means if I know the value of the function at some point x, I can find the value at -x by just taking the negative of the original value. So, if I can express f(2) in terms of f(-2), which is given by the provided expression, I can find the value.Let me write that down. Since f is odd, f(2) = -f(-2). That makes sense because if I plug in x = 2 into the odd function property, I get f(2) = -f(-2). So, I need to compute f(-2) first and then take its negative.Given that x = -2 is in the interval (-∞, 0), I can use the provided expression for f(x). So, f(-2) = 2*(-2)³ + (-2)². Let me compute that step by step.First, compute (-2)³. That's (-2)*(-2)*(-2). Let's see: (-2)*(-2) is 4, and then 4*(-2) is -8. So, (-2)³ = -8.Next, compute (-2)². That's (-2)*(-2), which is 4.So, plugging these back into the expression: f(-2) = 2*(-8) + 4.Calculating 2*(-8) gives -16, and then adding 4 gives -16 + 4 = -12. So, f(-2) is -12.Now, going back to the odd function property, f(2) = -f(-2). Since f(-2) is -12, f(2) = -(-12) = 12.Wait, let me double-check my calculations to make sure I didn't make a mistake. So, f(-2) is 2*(-2)^3 + (-2)^2. Let's compute each term:First term: 2*(-2)^3. As I did before, (-2)^3 is -8, so 2*(-8) is -16.Second term: (-2)^2 is 4.Adding them together: -16 + 4 is indeed -12. So, f(-2) is -12, and thus f(2) is 12.Just to make sure I didn't make any sign errors, let me go through it again.f(-2) = 2*(-2)^3 + (-2)^2.Compute (-2)^3: (-2)*(-2) = 4, then 4*(-2) = -8. So, 2*(-8) = -16.Compute (-2)^2: (-2)*(-2) = 4.Add them: -16 + 4 = -12. So, f(-2) = -12.Therefore, f(2) = -f(-2) = -(-12) = 12.Hmm, seems consistent. I don't think I made any mistakes here. So, f(2) should be 12.Wait a second, just to be thorough, let me recall the definition of an odd function. It's symmetric about the origin, meaning if you rotate the graph 180 degrees around the origin, it looks the same. So, if f(-x) = -f(x), then plugging in x = 2, we have f(-2) = -f(2). So, f(2) = -f(-2). Which is exactly what I used. So, that part is correct.Another way to think about it is if I have a point (a, b) on the graph of f(x), then the point (-a, -b) should also be on the graph. So, since f(-2) = -12, then f(2) should be 12. That makes sense.Just to visualize, if I plot f(x) for x < 0, it's 2x³ + x². Let me think about what that looks like. For negative x, x³ is negative, so 2x³ is negative. x² is positive, so f(x) is the sum of a negative and a positive term. Depending on the magnitude, it could be positive or negative.But in this case, when x = -2, f(-2) is -12, which is negative. So, the point (-2, -12) is on the graph. Therefore, the point (2, 12) should be on the graph as well because of the odd function property.So, all of this seems to confirm that f(2) is indeed 12.I think I'm confident with this answer. I don't see any mistakes in my reasoning or calculations.**Final Answer**The value of mathrm{f}(2) is boxed{12}.

answer：To determine the new average, I start with the given information that the average of ( x_1 ), ( x_2 ), and ( x_3 ) is 5. This means the sum of these three numbers is ( 3 times 5 = 15 ).Next, I add 2 to each of the numbers, resulting in ( x_1 + 2 ), ( x_2 + 2 ), and ( x_3 + 2 ). The sum of the new set of numbers is ( 15 + 6 = 21 ).Finally, to find the new average, I divide the total sum by the number of data points, which is 3. Therefore, the new average is ( frac{21}{3} = 7 ).

answer：First, I need to determine how much Marge has left after paying taxes. She won 12,000 and paid half of that for taxes, which is 6,000. This leaves her with 6,000.Next, she puts a third of the remaining 6,000 towards her student loans. A third of 6,000 is 2,000, so she has 4,000 left after paying her loans.Then, Marge decides to put 1,000 into savings. This reduces her remaining money to 3,000.After that, she invests a fifth of the 1,000 she saved into the stock market. A fifth of 1,000 is 200, so she has 800 left in her savings.Finally, to find out how much she has left for fun, I add the remaining 3,000 to the 800 left in savings, totaling 3,800.

answer：Okay, so I have this problem about systematic sampling. Let me try to understand what it's asking. There are 63 students in a class, each with a unique ID number. We need to select a sample of size 7 using systematic sampling. It's given that student ID number 11 is included in the sample, and we need to find the highest student ID number in the sample.First, let me recall what systematic sampling is. From what I remember, systematic sampling is a method where you select every k-th individual from a population. The value of k is determined by dividing the population size by the desired sample size. So, in this case, the population size N is 63, and the sample size n is 7. Therefore, k should be 63 divided by 7, which is 9. So, k is 9.That means we'll select every 9th student from the list. But how does this work exactly? I think the process is: you randomly choose a starting point between 1 and k, and then you select every k-th individual after that. For example, if the starting point is 3, then the sample would be 3, 12, 21, 30, 39, 48, 57.But in this problem, it's given that student ID 11 is included in the sample. So, that must mean that 11 is one of the selected IDs. Therefore, we need to figure out what the starting point is such that 11 is included in the sample.Wait, but if k is 9, then the starting point has to be such that when we add multiples of 9, we get 11. Let me think. If the starting point is s, then the sample would be s, s+9, s+18, s+27, s+36, s+45, s+54. So, one of these has to be 11.So, let's write that equation: s + 9*m = 11, where m is an integer between 0 and 6 (since the sample size is 7). So, s = 11 - 9*m. Since s has to be between 1 and 9 (because k is 9), let's see what m can be.If m=0, s=11, which is more than 9, so that's not possible. If m=1, s=11-9=2. That works because 2 is between 1 and 9. If m=2, s=11-18=-7, which is negative, so that's invalid. So, the only possible starting point is 2.Therefore, the starting point is 2, and the sample would be 2, 11, 20, 29, 38, 47, 56.Wait, let me check: 2, then 2+9=11, 11+9=20, 20+9=29, 29+9=38, 38+9=47, 47+9=56. Yes, that's 7 numbers, and 56 is the last one.So, the highest student ID number in the sample is 56. Therefore, the answer should be 56.But let me double-check if I did everything correctly. So, systematic sampling with k=9, starting point s=2, because 11 is in the sample. So, the sequence is 2, 11, 20, 29, 38, 47, 56. That seems right.Wait, another thought: sometimes in systematic sampling, if the starting point plus the interval exceeds the population size, you wrap around. But in this case, 56 is less than 63, so we don't have to worry about wrapping around. So, 56 is indeed the highest ID in the sample.Alternatively, if the starting point was different, say, if s was 11, but wait, s has to be less than or equal to k, which is 9. So, s can't be 11. So, the only way 11 is included is if the starting point is 2, because 2 + 9*1=11.Therefore, yeah, 56 is the highest ID in the sample.**Final Answer**The highest student ID number in the sample is boxed{56}.

answer：First, I need to determine the price per ton of organic fertilizers A and B. Let’s denote the price per ton of fertilizer B as ( x ) dollars. Since fertilizer A is 100 more expensive per ton than fertilizer B, the price per ton of fertilizer A would be ( x + 100 ) dollars.Given that purchasing 2 tons of fertilizer A and 1 ton of fertilizer B costs a total of 1700, I can set up the following equation:[2(x + 100) + x = 1700]Simplifying this equation:[2x + 200 + x = 1700][3x + 200 = 1700][3x = 1500][x = 500]So, the price per ton of fertilizer B is 500, and the price per ton of fertilizer A is ( 500 + 100 = 600 ) dollars.Next, to find out the maximum tons of fertilizer A Xiao Jiao can purchase while staying within a 5600 budget for a total of 10 tons, I'll let ( y ) represent the tons of fertilizer A. Consequently, the tons of fertilizer B would be ( 10 - y ).The total cost equation is:[600y + 500(10 - y) leq 5600]Simplifying:[600y + 5000 - 500y leq 5600][100y + 5000 leq 5600][100y leq 600][y leq 6]Therefore, Xiao Jiao can purchase at most 6 tons of fertilizer A.

answer：First, Matilda has 20 chocolate bars to share with her 4 sisters, making a total of 5 people. Dividing 20 by 5 gives each person 4 chocolate bars.Next, each sister, including Matilda, gives up half of their chocolate bars to their father. Half of 4 is 2, so each sister gives 2 bars to their father. With 5 sisters, the father receives a total of 10 chocolate bars.Then, the father gives 3 chocolate bars to their mother, leaving him with 7 bars. After eating 2 bars, the father has 5 chocolate bars remaining.