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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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Chapter
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BuyFindarrow_forward

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 3 cos x

(a)

To determine

To calculate: The interval where the first provided function f(x)=sinx3cosx is increasing or decreasing in the domain [0,2π].

Explanation

Given:

The function f(x)=sinx3cosx.

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)=cosx+3sinx

Equate this to zero to obtain the critical points.

cosx+3sinx=0cosx=3sinxtanx=13x=5π6,11π6

These critical points give three intervals:

(0,5π6),(5π6,11π6),(11π6,2π)

(b)

To determine

To calculate: The relative extrema of the function f(x)=sinx3cosx using the first derivative test.

(c)

To determine

To graph: The function f(x)=sinx3cosx and find the relative extrema.

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