Final answer:
To find the absolute maximum for the function f(x) = x^2 e^{-x}, we need to evaluate the function at the critical points and endpoints of the interval. The absolute maximum for the function on the interval [-1,3] is at x = 3.
Step-by-step explanation:
To find the absolute maximum for the function f(x) = x^2 e^{-x} on the interval [-1,3], we need to evaluate the function at the critical points and endpoints of the interval. To do this, we first find the derivative of the function, which is f'(x) = (2x - x^2)e^{-x}. Next, we find the critical points by setting the derivative equal to zero and solving for x. The critical point is x = 1. Finally, we evaluate the function at the critical point and endpoints to find the maximum value. Substituting x = -1, 1, and 3 into the function, we find that the maximum value occurs at x = 3. Therefore, the absolute maximum for the function is at x = 3.