   Chapter 3.1, Problem 56E

Chapter
Section
Textbook Problem

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

To determine

To find:

The absolute maximum and absolute minimum values of f(x) on the given interval.

Explanation

1) Concept:

Use Closed Interval Method to find the absolute maximum and minimum values of f(x).

The Closed Interval Method:

To find the absolute maximum and minimum values of a continuous function f on a closed interval a, b:

i. Find the values of f at the critical numbers of f in a, b.

ii. Find the values of f at the end points of the interval .

iii. The largest of the values from step (i) and (ii) is the absolute maximum value; the smallest of these values is the absolute minimum value.

2) Given:

ft=t+cott2, π4,7π4

3) Calculation:

Since f(t) is continuous on π4,7π4, use the Closed Interval Method.

Differentiate f(t) with respect to t, and then find the values of t, where f't=0 and f't doesn’t exist.

By using the chain rule of derivative,

f't= 1-12cosec2t2

Substitute  f't =0, and solve for t.

0=1-12cosec2t2

1=12cosec2t2

cosec2t2=2

cosect2= ±2

cosect2= 2

t2= cosec-12

t2=π4 or 3π4

t=π2 or 3π2 ….both lie in the interval π4,7π4

cosect2= -2

t2= cosec-1-2

t2=-π4 or 5π4

t=-π2 or 5π2 which does not lie in π4,7π4

Since f't exists for all t,t=π2 or3π2 are the critical point of ft

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In Exercises 69-74, rationalize the numerator. 72. 2x3y3

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### True or False: is a convergent series.

Study Guide for Stewart's Multivariable Calculus, 8th

#### Sometimes, Always, or Never: If f(x) is continuous on [a, b], then abf(x) dx exists.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 