`f(x) = 5 + 54x - 2x^3, [0, 4]` Find the absolute maximum and minimum values of f on the given interval

Textbook Question

Chapter 4, 4.1 - Problem 48 - Calculus: Early Transcendentals (7th Edition, James Stewart).
See all solutions for this textbook.

1 Answer | Add Yours

mathace's profile pic

mathace | (Level 3) Assistant Educator

Posted on

Given: `f(x)=5+54x-2x^3,[0,4]`

Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s).

`f'(x)=54-6x^2=0`

`54=6x^2`

`9=x^2`

`x=3,x=-3`

The critical value is x=3. The value x=-3 is not in the given interval [0, 4]. Therefore x=-3 will not be used to find the absolute maximum or absolute minimum value. Plug in the critical value x=3 and the endpoints of the interval [0, 4] into the original f(x) function. 

f(0)=5

f(3)=113

f(4)=93

Examine the f(x) value to determine the absolute maximum and absolute minimum. 

The absolute maximum occurs at the point (3, 113).

The absolute minimum occurs at the point (0, 5).

We’ve answered 318,915 questions. We can answer yours, too.

Ask a question