   Chapter 3.1, Problem 46E

Chapter
Section
Textbook Problem

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

To determine

To find:

The absolute maximum and absolute minimum values of f(x) on the given interval.

Explanation

1) Concept:

Use the Closed Interval Method to find the absolute maximum and minimum values of f(x).

The Closed Interval Method:

To find the absolute maximum and minimum values of a continuous function f on a closed interval a, b:

i. Find the values of f at the critical numbers of f in a, b.

ii. Find the values of f at the endpoints of the interval .

iii. The largest of the values from step (i) and (ii) is the absolute maximum value; the smallest of these values is the absolute minimum value.

2) Given:

fx=5+54x-2x3, [0, 4]

3) Calculation:

Since f(x) is continuous on [0, 4], use the Closed Interval Method.

Differentiate f(x) with respect to x, and then find the values of x where f'x=0 and f'x doesn’t exist.

By using the power rule of derivative,

f'x=54-6x2

Substitute f'x =0, and solve for x.

0=6(9-x2)

Hence, x=3, -3

x=-3 does not lie in the given interval [0, 4].

Since f'x exists for all x, then x=3 is the only critical point of fx

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