Question

Determine the absolute maximum and

absolute minimum of f(x) = x³ - 3x² + 1 on
the interval [-1/2,4].

Answer 1

• maximum: 17
• minimum: -3

Explanation:

You want the absolute extremes of the function f(x) = x³ -3x² +1 on the closed interval [-1/2, 4].

Extreme values

The extreme values will be either the turning points of the function or the ends of the interval. We must examine all of those. A graphing calculator is helpful for evaluating the function at various points.

Interval ends

f(-1/2) = 1/8

f(4) = 17

Turning points

The turning points are found where the derivative is zero.

f'(x) = 3x² -6x = 3x(x -2)

This function is zero where the factors are 0, at x=0 and x=2.

f(0) = 1

f(2) = -3

The maximum value is found at the right end of the interval: f(4) = 17.

The minimum value is found at the right turning point: f(2) = -3.