sketch a graph of a function f with the following properties. f>0 and f>0, for x < -7 f>0 and f<0, for…

Explanation:

Step1: Recall derivative - function behavior

$f'>0$ means the function is increasing, $f'<0$ means it is decreasing. $f''>0$ means the function is concave - up, $f''<0$ means it is concave - down.

Step2: Analyze intervals

For $x < - 7$, $f'>0$ and $f''>0$, so the function is increasing and concave - up. For $-7<x < - 6$, $f'>0$ and $f''<0$, so the function is increasing and concave - down. For $-6<x < 2$, $f'>0$ and $f''>0$, so the function is increasing and concave - up. For $2<x < 4$, $f'>0$ and $f''>0$, so the function is increasing and concave - up. For $x>4$, $f'>0$ and $f''<0$, so the function is increasing and concave - down.

Step3: Match with graphs

By analyzing the increasing/decreasing and concave - up/concave - down behavior in each interval, we can match the function with the correct graph.

Answer:

(Without seeing the full - sized and detailed graphs, it's hard to give a specific letter answer. But the correct graph should have the increasing and concavity behavior as described above. If we assume we have analyzed the graphs properly based on the text description, we would choose the graph that matches the behavior in each interval). If we had to make a general statement, we need to look for a graph that has an increasing trend throughout (since $f'>0$ always) and changes concavity at $x=-7,x = - 6,x = 4$.)