Chapter 4.6, Problem 57E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

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

# Analyzing the Graph of a Function Using TechnologyIn Exercises 55-62, use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. y = cos x − 1 4 cos 2 x ,       0 ≤ x ≤ 2 π

To determine

To graph: The provided function y=cosx14cos2x where x lies between 0 and 2π with the help of a graphing utility and to obtain the extremas, inflection points and asymptotes for the function.

Explanation

Given:

y=cosxâˆ’14cos2x

Graph:

The graph of the function can be obtained using Maple.

Maple Command:

Maple Graph:

Now consider the provided function.

y=cosxâˆ’14cos2x

Cleary, the function is defined for all values of x andthus the function has no vertical asymptotes.

Now differentiate the provided function and equate the derivative to zero to obtain the critical points.

f'(x)=âˆ’sinx+12sin2xâˆ’sinx+12sin2x=0x=0,Ï€,2Ï€

Now obtain the value of the function at these points x=0,Ï€,2Ï€.

y(0)=cos0âˆ’14cos2(0)=0.75y(Ï€)=cosÏ€âˆ’14cos(2Ï€)=âˆ’1.25

And,

y(2Ï€)=cos(2Ï€)âˆ’14cos(4Ï€)=0.75

From the absolute extrema approach, the relative minima is (Ï€,âˆ’1.25).

Now differentiate the function twice and equate that to zero to get the inflection points.

f''(x)=âˆ’cosx+cos2xâˆ’cosx+cos2x=0x=2Ï€3,4Ï€3

Now obtain the value of the function at the inflection points

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