Final answer:
To determine the absolute extreme values of the function f(x) = 5sin²(x) on the interval [0, π], we need to find the maximum and minimum values of the function within that interval. The maximum value is 5 and it occurs at x = π/2, and the minimum value is 0 and it occurs at x = 0 and x = π.
Step-by-step explanation:
To determine the absolute extreme values of the function f(x) = 5sin²(x) on the interval [0, π], we need to find the maximum and minimum values of the function within that interval. First, we can find the critical points by taking the derivative of the function. The derivative of f(x) = 5sin²(x) is f'(x) = 10sin(x)cos(x). Setting f'(x) = 0, we get sin(x)cos(x) = 0. This occurs when sin(x) = 0 or cos(x) = 0. The critical points within the interval [0, π] are x = 0, x = π/2, and x = π.
Now we can test the values of these critical points and the endpoints of the interval to find the maximum and minimum values of the function. Evaluating the function at these points, we get f(0) = 0, f(π/2) = 5, f(π) = 0. Therefore, the maximum value of the function is 5 and it occurs at x = π/2, and the minimum value of the function is 0 and it occurs at x = 0 and x = π.