Answer
False
Work Step by Step
When $f'(c)=0$ is only means that the tangent line to f at $x =c$ is horizontal.
The function $f(x)=x^{3}$ has $f'(x)=2x^{2}$
$f'(0)=0$ but $f'(x)\gt 0$ and increasing for all other $x$. In this case $x=c$ is an inflection point and not a local maximum or minimum.
Hence, the given statement is false.
