Suppose $h(x)$ is a function such that $h(x)$ has exactly three critical point. Two of which are shown in the table below.
Assume that both $h(x)$ and $h^{\prime}(x)$ are differentiable on $(-\infty, \infty)$
$$
\begin{array}{|c|c|c|c|c|}
\hline x & 0 & 3 & 5 & 7 \\
\hline h(x) & 2 & ? & 4 & 4 \\
\hline h^{\prime}(x) & -1 & 0 & 0 & ? \\
\hline
\end{array}$$
Further using Lagrange mean value theorem in the interval $[5,7],$ we can determine the interval of the third critical point.
On which of the following intervals must $h(x)$ be increasing on the entire interval?
- $(0,3)$
- $(3,5)$
- $(5,6)$
- $(6,7)$
