Question

level 4: open ended questions

1. sketch the graph of a single function with a domain (-∞,∞) that has all of the following characteristics:

a. concave down and increasing on the interval (-∞,-4)
b. concave down and decreasing on the interval (-4,2)
c. concave up and decreasing on the interval (2,7)
d. concave up and increasing on the interval (7,∞)

Explanation:

Step1: Recall function - behavior rules

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

Step2: Sketch on $(-\infty,-4)$

On the interval $(-\infty,-4)$, since the function is concave down ($f''(x)<0$) and increasing ($f'(x)>0$), the graph has a positive slope and is bending downwards. Start the graph with an increasing curve that is concave down.

Step3: Sketch on $(-4,2)$

On the interval $(-4,2)$, as the function is concave down ($f''(x)<0$) and decreasing ($f'(x)<0$), the graph has a negative slope and is bending downwards. Connect the previous part with a decreasing curve that is concave down.

Step4: Sketch on $(2,7)$

On the interval $(2,7)$, because the function is concave up ($f''(x)>0$) and decreasing ($f'(x)<0$), the graph has a negative slope and is bending upwards. Connect the previous part with a decreasing curve that is concave up.

Step5: Sketch on $(7,\infty)$

On the interval $(7,\infty)$, since the function is concave up ($f''(x)>0$) and increasing ($f'(x)>0$), the graph has a positive slope and is bending upwards. Connect the previous part with an increasing curve that is concave up.

Answer:

A hand - sketched graph following the above - described behavior on the given intervals on the provided grid.