   Chapter 4.2, Problem 7E ### 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

# Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.f(x) = sin(x/2), [π/2, 3π/2]

To determine

To verify: The function f(x)=sin(x2) on interval [π2,3π2] that satisfies the conditions of Rolle’s Theorem to find All the numbers of c of the function f(x)=sin(x2) that satisfy the conclusion of Rolle’s Theorem.

Explanation

Given:

The function f(x)=sin(x2) is a sine function.

Rolle’s Theorem:

“Let a function f satisfies the following conditions,

1. A function f is continuous on the closed interval [a,b] .

2. A function f is differentiable on the open interval (a,b) .

3. f(a)=f(b)

Then there is a number c in open interval (a,b) such that f(c)=0

Verification:

1. Since the sine function f(x)=sin(x2) is continuous, the function f(x) is continuous on the closed interval [π2,3π2] .

2. Since the sine function f(x)=sin(x2) is differentiable everywhere, the function f(x) is differentiable on the open interval (π2,3π2) .

3. To satisfy the third condition of the theorem, f(a) must be equal to f(b) .

From the given interval, it is observed that a=π2 and b=3π2 .

Substitute π2 for x in f(x) ,

f(x)=sin((π2)2)=sin(π4)=12[sin(π4)=12]

Substitute 3π2 for x in f(x) ,

f(x)=sin((3π2)2)=sin(3π4)=22

### 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

#### Simplify the expressions in Exercises 97106. x1/2yx2y3/2

Finite Mathematics and Applied Calculus (MindTap Course List)

#### A=12bh,b=40,andh=15,findA

Elementary Technical Mathematics

#### In Problems 19-44, factor completely. 40.

Mathematical Applications for the Management, Life, and Social Sciences

#### ∫(x − cos x)dx = 1 − sin x + C 1 + sin x + C

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

#### Graph each function. fx=ex-3

College Algebra (MindTap Course List) 