Chapter 4.3, Problem 58E

### 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 ) = sin x cos x + 5

(a)

To determine

To calculate: The interval where the first provided function is increasing or decreasing in the domain [0,2π].

Explanation

Given:

The function f(x)=sinxcosx+5.

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)=âˆ’sin2x+cos2x=cos2x

Equate this to zero to obtain the critical points.

cos2x=02x=Ï€2,3Ï€2,5Ï€2,7Ï€2x=Ï€4,3Ï€4,5Ï€4,7Ï€4

These critical points give five intervals (0,Ï€4),(Ï€4,3Ï€4),(3Ï€4,5Ï€4),(5Ï€4,7Ï€4),(7Ï€4,2Ï€)

(b)

To determine

To calculate: The relative extrema of the function f(x)=sinxcosx+5 using the first derivative test.

(c)

To determine

To graph: The function f(x)=sinxcosx+5 and find the relative extrema.

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