   Chapter 4.1, Problem 58E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the absolute maximum and absolute minimum values of f on the given interval.f(t) = t + cot (t/2), [π/4, 7π/4]

To determine

To find: The absolute maximum and absolute minimum of the given function.

Explanation

Given:

The function f(t)=t+cot(t2) in the interval [π4,7π4] .

Calculation:

Obtain the first derivative of the given function.

f(t)=ddt(t+cot(t2))=ddt(t)+ddt(cot(t2))=1+(csc2(t2))ddt(t2)(ddt(cott=csc2t))=1csc2(t2)(12)

Set f(t)=0 and obtain the critical numbers.

1csc2(t2)(12)=0csc2(t2)=2csc(t2)=±2t2=π4,t2=3π4

Hence, the critical numbers are  t=π2,t=3π2

Here, both the critical numbers are lies in the given interval [π4,7π4] .

Apply the extreme values of the given interval and the critical numbers in f(t) .

Substitute t=π4 in f(t) ,

f(π4)=π4+cot(π42)=π4+cot(π8)=π4+cot(π8)3

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