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 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

Answer to Problem 14RQ

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.

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