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

Question

f(x) = x³ − 9x² + 9,    [−4, 8]

Absolute Minimum Value?

Absolute Maximum Value?

Answer 1

There are three places to check:

1) local minimums or maximums (located within the interval) found by setting the derivative equal to zero

2) places where the derivative does not exist (located within the interval).

3) the endpoints of the interval.

Look at all three and find the value that is largest (absolute max) and smallest (absolute min)

Start with #1. Take the derivative, set it equal to zero, and solve:

f(x) = x³ − 9x² + 9

f '(x) = 3x² - 18x (using the power rule)

3x² - 18x = 0

3x(x - 6) = 0

x = 0 and x = 6 and both of these are on the interval.

Find the function value of each:

f(0) = 9

f(6) = -99

Step 2 - look for places the derivative does not exist

There are none

Step 3 - Look at the endpoints:

f(-4) = -199

f(8) = -55

So the absolute max is 9 which occurs at x = 0 and the absolute min is -199 which occurs at x = -4