Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

4) 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 \(f'\left( c \right) = 0\).

Short answer

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 \(f'\left( c \right) = 0\), is true using Rolle’s theorem.

Step-by-step solution

Given

Givenf is differentiable.

Such that,\(f\left( { - 1} \right) = f\left( 1 \right)\)

f is differentiable on

f is continuous on and

( given )

Apply Rolle’s theorem

Now by Rolle’s theorem,

\(\exists c \in \left( { - 1,1} \right)\)

Such that \(f'\left( c \right) = 0\)

Since \(c \in \left( { - 1,1} \right)\)

\(\therefore \left| c \right| < 1\)

And \(f'\left( c \right) = 0\) ( by above )

Hence our statement is true.