Calculus Question Help

Question

Find the absolute maximum and absolute minimum values of f on the given interval.

f(x) = x³ − 6x² + 9x + 7,    [−1, 8]

Answer 1

We need to evaluate at critical points and endpoints of domain.

Critical points: Find the zeroes of f'(x) = 3x² - 12x + 9 = 3(x-3)(x-1). So critical values of x are 1 and 3

f(-1) = -1 - 6*1 + 9*(-1) + 7 = -9

f(1) = 1 - 6*1 + 9*1 + 7 = 11

f(3) = 27 - 6*9 + 9*3 + 7 = 7

f(8) = 512 - 6*64 + 9*8 + 7 = 207

Absolute minimum value is -9, assumed when x = -1

Absolute maximum value is 207, assumed when x = 8