   Chapter 4.4, Problem 45E 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

Using the Second Derivative Test InExercises 37-58, find all relative extrema of the function. Use the Second Derivative Test where applicable. f ( x ) = cos x − x ,     [ 0 , 4 π ]

To determine

To calculate: The relative extrema of the provided function f(x)=cosxx using the second derivative test.

Explanation

Given:

The function f(x)=cosxx over the interval [0,4π].

Formula used:

For a function f that is twice differentiable on an open interval I,

If f'(c)=0 for some c, and,

If f''(c)>0 the function f has relative minima at c

If f''(c)<0 the function f has relative maxima at c.

If f''(c)=0, the test fails.

Calculation:

First differentiate the provided function,

f'(x)=sinx1

Equate the first derivative to zero to obtain the critical points.

sinx1=0sinx=1x=π2,5π2

Now differentiate the function for the second time and replace this for x:

f''(π2)=cos(π2)=0

This implies that the test fails at this point.

Also,

f''(5π2)=cos(5π2)=0

This implies that the test fails at this point

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