Chapter 4.3, Problem 61E

### 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

# Applying the First Derivative Test In Exercises 57-62, consider the function on the interval ( 0 , 2 π ) . (a) Find the open intervals on which the function is increasing or decreasing.(b) Apply the First Derivative Test to identify all relative extrema, (c) Use a graphing utility to confirm your results. f ( x ) = cos 2 ( 2 x )

(a)

To determine

To calculate: The interval where the first provided function f(x)=cos2(2x) is increasing or decreasing in the domain [0,2π].

Explanation

Given:

The function f(x)=cos2(2x).

Formula used:

The function f said to have a critical point at c if:

f'(c)=0

For any function g(x) that is differentiable in the interval (a,b) and continuous in the interval [a,b], the function g(x) can be classified as an increasing or a decreasing function based on the following conditions:

If d(g(x))dx>0 for the interval (a,b), the function g(x) is said to be increasing in the interval [a,b].

If d(g(x))dx<0 for the interval (a,b), the function g(x) is said to be decreasing in the interval [a,b].

Calculation:

Differentiate the provided function.

f'(x)=2cos2x(âˆ’sin2x)(2)=âˆ’2sin4x

Equate this to zero to obtain the critical points.

âˆ’2sin4x=0sin4x=04x=Ï€,2Ï€,3Ï€,4Ï€,5Ï€,6Ï€,7Ï€x=Ï€4,Ï€2,3Ï€4,Ï€,5Ï€4,3Ï€2,7Ï€4

These critical points give eight intervals:

(0,Ï€4),(Ï€4,Ï€2),(Ï€2,3Ï€4),(3Ï€4,Ï€),(Ï€,5Ï€4),(5Ï€4,3Ï€2)(3Ï€2,7Ï€4),(7Ï€4,2Ï€).

Take a test point in the interval and check the sign of the derivative.

Let Ï€12âˆˆ(0,Ï€4).

f'(Ï€12)=âˆ’2sin(4(Ï€12))=âˆ’3<0

The function is decreasing in this interval

(b)

To determine

To calculate: The relative extrema of the function f(x)=cos2(2x) using the first derivative test.

(c)

To determine

To graph: The function f(x)=cos2(2x) and find the relative extrema.

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