Final answer:
To find the absolute minimum value of the function f(x) = xe^{-3x}, we need to find the critical points of the function and evaluate the function at those points. The critical points occur when the derivative of the function is equal to zero or does not exist. Taking the derivative of f(x) with respect to x, we find the critical point x = 1/3 and evaluating the function at x = 1/3 gives us the absolute minimum value.
Step-by-step explanation:
The function f(x) = xe-3x, for 0 ≤ x ≤ 2, has an absolute minimum value. To find the absolute minimum, we need to find the critical points of the function and evaluate the function at those points. The critical points occur when the derivative of the function is equal to zero or does not exist. Taking the derivative of f(x) with respect to x, we get f'(x) = e-3x(1 - 3x).
Setting f'(x) = 0, we have e-3x(1 - 3x) = 0. Either e-3x = 0, which is not possible, or 1 - 3x = 0. Thus, 3x = 1 and x = 1/3. Evaluating f(x) at x = 1/3, we have f(1/3) = (1/3)e-3/3 = (1/3)e-1. This is the absolute minimum value of the function.