# Find the increasing interval for the given function.

### 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 34E
To determine

## Find the increasing interval for the given function.

Expert Solution

The function is concave downwards on (,π2) and (π4,π2) .

### Explanation of Solution

Given:

The given functionis f(x)=2xtanx , π2<x<π2 .

Calculation:

f(x)=2xtanx

Apply difference rule.

(fg)'=f'g'

f'(x)=ddx(2x)ddx(tanx)

Use derivative rule ddx(xn)=nxn1 and ddx(tanx)=sec2x

f'(x)=2sec2x

Solve for f'(x)=0

2sec2x=02sec2x2=02sec2x=2sec2x=21cos2x=2cos2x=12cosx=±12

It is in the interval (π2,π2) at π4,π4 .

Find the second derivative.

f"(x)=ddx(2)ddx(sec2x)f"(x)=02secxsecxtanxf"(x)=2sec2xtanx

Solve for f''(x)=0

2sec2xtanx=0sec2xtanx=0sec2x=0(nosolution)tanx=0x=nπ,n=1,2,3

Substitute x=π4 .

f''(π4)=2sec2(π4)tan(π4)=22(1)=4(positive)

Substitute x=π4 .

f''(π4)=2sec2(π4)tan(π4)=221=4(negative)

Hence thefunction is concave downwards on (,π2) and (π4,π2) .

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