Answer
See the explanation below.
Work Step by Step
Let us consider that a function $f(x)$ is continuous on a closed interval $[a,b]$ and located at $x=c$.
a. When $f' (c) =0$ and $f'' (c) \lt 0$ on an interval $[a,b]$, then the function $f(x)$ has a local maximum at the point $x=c$.
b. When $f' (c) =0$ and $f'' (c) \gt 0$ on an interval $[a,b]$, then the function $f(x)$ has a local minimum at the point $x=c$.
c. When $f' (c) =0$ and $f'' (c) =0$ on an interval $[a,b]$, then the function $f(x)$ fails the Second Derivative Test and the function $f(x)$ might have a local minimum and a local maximum or neither at $x=c$.
