Chapter 4 - Questions to Guide Your Review - Page 275: 10

See the explanation below.

Suppose that a function $f(x)$ is continuous on a closed interval $[m,n]$ and located at $x=c$. 1. When $f' (c) =0$ and $f'' (c) \lt 0$ on an interval $[m,n]$, then the function $f(x)$ attains a local maximum at point $x=c$. 2. When $f' (c) =0$ and $f'' (c) \gt 0$ on an interval $[m,n]$, then the function $f(x)$ attains a local minimum at point $x=c$. 3. When $f' (c) =0$ and $f'' (c) =0$ on an interval $[m,n]$, then the function $f(x)$ fails the Second derivative Test and the function $f(x)$ may have either a local minimum or local maximum or neither at the point $x=c$. We apply the Second derivative Test during a sketch of the curve to determine the local extrema (local minimum or local maximum ) of a function $f(x)$.