Answer
False statement
Work Step by Step
Consider $f(x)=x^{3/2}$.
$f'(x)=\frac{3}{2}\sqrt{x}\implies f'(0)=0$. $f'(c)$ exists at $c=0$.
$f''(x)=\frac{3}{4\sqrt{x}}$
Note that $f''(0)$ doesn't exist.
Hence even if $f'(c)$ exists we cannot conclude that $f''(c)$ exists.
