5 multiple choice 1 point the figure above shows the graph of f, the derivative of the function f, on the…

Answer:

C. Three

Explanation:

Step1: Recall inflection - point condition

A point of inflection of the function $y = h(x)$ occurs where $h''(x)=0$ and $h''(x)$ changes sign. Given $h'(x)=2f'(x)$, then $h''(x) = 2f''(x)$. So, points of inflection of $h(x)$ occur where $f''(x)=0$ and $f''(x)$ changes sign.

Step2: Identify $f''(x) = 0$ from $f'(x)$ graph

The second - derivative $f''(x)$ is the derivative of $f'(x)$. The derivative of a function is zero at the local maxima and minima of the function. The local maxima and minima of $y = f'(x)$ occur where the slope of $y = f'(x)$ is zero.

Step3: Count local extrema of $f'(x)$

Looking at the graph of $y = f'(x)$, we can see that the slope of $y = f'(x)$ is zero at three points in the interval $[-3,6]$. These are the points where the graph of $y = f'(x)$ has a local maximum or minimum. At these points, $f''(x)=0$ and $f''(x)$ changes sign. So, the graph of $h(x)$ has three points of inflection in the interval $[-3,6]$.