Whether the statement, “If f is continuous on [ − 1 , 1 ] and f ( − 1 ) = 4 and f ( 1 ) = 3 , then there exists a number r such that | r | &lt; 1 and f ( r ) = π ” is true or false.

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 2, Problem 14RQ
To determine

Whether the statement, “If f is continuous on [−1, 1] and f(−1)=4 and f(1)=3, then there exists a number r such that |r|<1 and f(r)=π” is true or false.

Expert Solution

The statement is false.

Explanation of Solution

Theorem used: The Intermediate value Theorem

Suppose that if f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a)f(b). Then there exists a number c in (a, b) such that f(c)=N.

Reason:

Suppose f is continuous on [1,1] and f(1)=4 and f(1)=3.

Let N=π lies between 3 and 4. That is, f(1)<N<f(1).

Then by Intermediate value theorem states that, there is a number r in the interval (−1, 1) such that f(r)=N.

Thus, there is a number r such that 1<r<1 (or |r|<1) and f(r)=π.

Therefore, the given statement is true.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!