CPM Homework Help : CALC Problem 3-117

Sketch a function $f(x)$ for which the following is true about its slope function $f^\prime(x)$ .

For $x < −3$ , $f^\prime(x) > 0$ and the slope is increasing.
For $−3 < x < −1$ , $f^\prime(x) > 0$ and the slope is decreasing.
At $x = −1$ , $f^\prime(x)= 0$ .
For $x > −1$ , $f^\prime(x) < 0$ and the derivative is decreasing.

Hint:

Be careful! The clues give information about $f^\prime(x)$ , but you are asked to sketch $f(x)$ .

Hint 1:

Translate: When $x$ is greater than $−3$ , $f(x)$ is increasing and concave up.

Hint 2:

Translate: When $x$ is in between $−3$ and $−1$ , $f(x)$ is increasing and concave down.

Hint 3:

Translate: When $x$ is exactly $−1$ , $f(x)$ has reached a local maximum or minimum.

Hint 4:

Translate: When $x$ is greater than $−1$ , $f(x)$ is decreasing and concave down.

Graph:

Remember that this graph is based on a slope statement. Without information about the exact $y$ -values of $f(x)$ ,the entire sketch can be shifted vertically up or vertically down.