# The values of x in which the graph of f ( x ) have a horizontal tangent. ### 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 3.3, Problem 32E
To determine

## To find: The values of x in which the graph of f(x) have a horizontal tangent.

Expert Solution

The values of x in which the graph of f(x) have a horizontal tangent is x=π4+nπ_, where n is an integer.

### Explanation of Solution

Given:

The function is f(x)=excosx.

Derivative rule:

Product Rule: If f(x). and g(x) are both differentiable function, then

ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)] (1)

Calculation:

Obtain the derivative of f(x).

Apply the product rule (1) and simplify further,

ddx[excosx]=exddx[cosx]+cosxddx[ex]=ex[sinx]+cosx[ex]=exsinx+excosx=ex(sinx+cosx)

Thus, the derivative of   f(x)=excosx is ex(sinx+cosx).

Note that, the function f(x) has horizontal tangent when f(x)=0.

ex(sinx+cosx)=0sinx+cosx=0cosx=sinx

Since the values of sinπ4 and cosπ4 are equal, the general case is x=π4+nπ.

Therefore, the values of x in which the graph of f(x) have a horizontal tangent is x=π4+nπ_, where n is an integer.

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