   Chapter 4.3, Problem 14E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f.(c) Find the intervals of concavity and the inflection points.f(x) = cos2x − 2 sin x, 0 ≤ x 2π

(a)

To determine

To find: The intervals on which function f(x)=cos2x2sinx on [0,2π] is increasing or decreasing.

Explanation

Given:

The function is f(x)=cos2x2sinx.

Calculation:

Obtain the derivative of f(x).

f(x)=ddx(cos2x2sinx)=2cosx(sinx)2(cosx)=2cosxsinx2cosx=2cosx(sinx+1)

Set f(x)=0 and obtain the critical points.

2cosx(sinx+1)=0cosx=0,sinx+1=0x=0,sinx=1x=π2,x=3π2

Hence, the critical points occurs at x=π2, x=3π2 and in the interval notation (0,π2)(π2,3π2)(3π2,2π).

Determine the nature of f(x) as follows.

 Interval f'(x)=−2cosx(sinx+1) f(x) 0

(b)

To determine

To find: The local maximum and minimum values of f(x)=cos2x2sinx on [0,2π].

(c)

To determine

To find: The intervals of concavity and the inflection points of f(x)=cos2x2sinx

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