Chapter 4, Problem 4RQ

To determine

Whether the statement, if f is differentiable and $f\left(-1\right)=f\left(1\right)$ then there is a number c such that $\left|c\right|<1$ and $f^{\prime }\left(c\right)=0$ is true or false.

Answer to Problem 4RQ

The given statement is true.

Explanation of Solution

Reason:

The function f is a continuous function as f is differentiable function.

Also it is given that, $f\left(-1\right)=f\left(1\right)$.

So, the local maximum or a local minimum occurs between x = −1 and x = 1.

There must be a point c such that, at the point c the value of the derivative is zero.

That is there exist a point c such that, $\left|c\right|<1$ and $f^{\prime }\left(c\right)=0$.

Therefore, the given statement is true.