3. (1 point) the graph of h is shown below. sketch the graph of h .

Explanation:

Step1: Recall second - derivative rules

The second - derivative $h''(x)$ tells about the concavity of $h(x)$. If $h''(x)>0$, the function $h(x)$ is concave up; if $h''(x)<0$, the function $h(x)$ is concave down.

Step2: Analyze intervals of concavity

• For $x\in(-2,0)$, the function $h(x)$ is concave down. So $h''(x)<0$ on this interval.
• For $x\in(0,3)$, the function $h(x)$ is concave up. So $h''(x)>0$ on this interval.
• For $x\in(3,5)$, the function $h(x)$ is concave up. So $h''(x)>0$ on this interval.

Step3: Identify inflection points

The inflection points of $h(x)$ occur where the concavity changes. Here, the inflection point is at $x = 0$ and $x=3$ since the concavity of $h(x)$ changes at these points. At inflection points, $h''(x)=0$.

Answer:

Sketch a graph of $h''(x)$ such that it is negative on $(-2,0)$, crosses the $x$ - axis at $x = 0$, is positive on $(0,3)$, crosses the $x$ - axis at $x = 3$, and is positive on $(3,5)$. The $y$ - values of $h''(x)$ can be estimated based on the rate of change of the concavity of $h(x)$. For example, the magnitude of $h''(x)$ can be larger where the concavity of $h(x)$ is more pronounced.