   Chapter 4.1, Problem 4E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Extrema on a Closed Interval Explain how to find the extrema of a continuous function on a closed interval [a, b]

To determine

The explanation of “The procedure to find the extrema of a continuous function on a closed interval [a,b].

Explanation

Steps to find the extrema of function are as below:

1. Find the critical numbers of the function.

2. Find the function values f(c) for each critical numberc.

3. Evaluate the function at the endpoints.

4. The largest value is the absolute maximum and the smallest value is the absolute minimum.

For example,

Consider the function,

f(x)=2x3+3x212x+4,[4,2]

Differentiate the function.

That is.,

f'(x)=6x2+6x12

Set f'(x)=0, to find the limit point.

That is.,

6x2+6x12=0x2+x2=0x2+2xx2=0(x1)(x+2)=0

This implies,

x=1 or -2

Evaluate the function at the critical points.

That is.,

f(1)=2(1)3+3(1)212(1)+4=2+312+4=3

And,

f(2)=2(2)3+3(2)212(2)+4=16+1224+4=24

Now,

Left end point is x=4

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