Question

1. sketch a single graph with the characteristics described below:

● concave up and decreasing on the interval (-∞, -2)
● concave up and increasing on the interval (-2, 3)
● concave down and increasing on the interval (3,6)
● concave down and decreasing on the interval (6, ∞)

Explanation:

Step1: Recall function - behavior rules

The sign of the first - derivative $f'(x)$ determines if a function is increasing ($f'(x)>0$) or decreasing ($f'(x)<0$), and the sign of the second - derivative $f''(x)$ determines if a function is concave up ($f''(x)>0$) or concave down ($f''(x)<0$).

Step2: Start sketching on $(-\infty,-2)$

On the interval $(-\infty,-2)$, since the function is concave up ($f''(x)>0$) and decreasing ($f'(x)<0$), the graph should be curving upwards and sloping downwards.

Step3: Sketch on $(-2,3)$

On the interval $(-2,3)$, as the function is concave up ($f''(x)>0$) and increasing ($f'(x)>0$), the graph should be curving upwards and sloping upwards.

Step4: Sketch on $(3,6)$

On the interval $(3,6)$, with the function being concave down ($f''(x)<0$) and increasing ($f'(x)>0$), the graph should be curving downwards and sloping upwards.

Step5: Sketch on $(6,\infty)$

On the interval $(6,\infty)$, since the function is concave down ($f''(x)<0$) and decreasing ($f'(x)<0$), the graph should be curving downwards and sloping downwards.

Answer:

A hand - sketched graph following the above - described behaviors on the given intervals. (It's not possible to actually draw the graph in this text - based format, but you can follow the steps above to draw it on the provided grid paper. Start with a decreasing and concave - up curve on $(-\infty,-2)$, then transition to an increasing and concave - up curve on $(-2,3)$, then to an increasing and concave - down curve on $(3,6)$, and finally to a decreasing and concave - down curve on $(6,\infty)$.)