BuyFind

Single Variable Calculus: Concepts...

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

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

Answer to Problem 34E

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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