Chapter 4.3, Problem 65ES

The accompanying figure shows the graph of the derivative of a function $h$ that is defined and continuous on the interval $\left(-\infty ,+\infty \right)$ . Assume that the graph of $h^{\prime }$ has a vertical asymptote at $x=3$ and that $\begin{array}{l}{h}^{\prime }\left(x\right)\to 0^{+}\text{ as }x\to -\infty \\ h^{\prime }\left(x\right)\to -\infty \text{ as }x\to +\infty \end{array}$

(a) What are the critical points for $h\left(x\right)$ ?

(b) Identify the intervals on which $h\left(x\right)$ is increasing.

(c) Identify the x-coordinates of relative extrema for $h\left(x\right)$ and classify each as a relative maximum or relative minimum.

(d) Estimate the x-coordinates of inflection points for $h\left(x\right)$ .