# The values of the number a for which the function f have no critical number.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 11P
To determine

## The values of the number a for which the function f have no critical number.

Expert Solution

### Explanation of Solution

Given information:

The function is,

f(x)=(a2a6)cos2x+(a2)x+cos1

Calculations:

Here the given function is,

f(x)=(a2a6)cos2x+(a2)x+cos1

Now to get the critical number, we have find the derivative of the function f and find its domain.

f(x)=(a2a6)cos2x+(a2)x+cos1f'(x)=2(a2a6)sin2x+(a2)

Now when f'(x)=0 we get,

2(a2+a6)sin2x+(a2)=02(a2+a6)sin2x=2a2sin2x[(a2+3a2a6)]=2a2sin2x[2(a+3)2(a+3)]=2a2sin2x[(a2)(a+3)]=2a2sin2x.(a+3)=1sin2x=12(a+3)Thus we can see that if a=3, then we can nat find a definite value.

Therefore if the value of a is 3 then, we cannot get a critical number.

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