Question

Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 3x^4 − 4x^3 − 12x^2 + 1, [−2, 3]

Answer 1

Find the stationary points.

f(x) = 3x ⁴ - 4x ³ - 12x ² + 1

f '(x) = 12x ³ - 12 x ² - 24x

Solve f '(x) = 0.

12x ³ - 12 x ² - 24x = 12x (x ² + x - 2) = 12x (x - 1) (x + 2) = 0

→   x = 0, x = 1, x = -2

Check the value of f at the stationary points.

f (0) = 1

f (1) = -12

f (-2) = 33

Check the value of f at the boundary of the domain.

f (3) = 28

(We've already checked f (-2).)

Then over [-2, 3], we have max(f ) = 33 and min(f ) = -12.