Final answer:
The absolute maximum is 12, which occurs at x = -1, and the absolute minimum is -25, which occurs at x = 4.
Step-by-step explanation:
To find the absolute maximum and absolute minimum of the function f(x) = x³ - 6x² + 7 on the interval -1 ≤ x ≤ 6, we can first find the critical points and then evaluate the function at the critical points and the endpoints of the interval.
To find the critical points, we take the derivative of f(x) and set it equal to zero.
The derivative of f(x) is f'(x) = 3x² - 12x. Setting f'(x) = 0, we get 3x² - 12x = 0. Factoring out 3x, we have 3x(x - 4) = 0. So the critical points are x = 0 and x = 4.
Next, we evaluate f(x) at the critical points and the endpoints of the interval.
f(-1) = (-1)³ - 6(-1)² + 7 = -1 + 6 + 7 = 12
f(6) = 6³ - 6(6)² + 7 = 216 - 216 + 7 = 7
f(0) = 0³ - 6(0)² + 7 = 7
f(4) = 4³ - 6(4)² + 7 = 64 - 96 + 7 = -25
So the absolute maximum is 12, which occurs at x = -1, and the absolute minimum is -25, which occurs at x = 4.