CPM Homework Help : APCALC Problem 3-117

Sketch a function $f$ for which the following statements are true about its slope function $f^\prime$ .

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.

Continuous curve, coming from left above x axis, opening up, changing concavity, then turning at about (negative 1, comma 3), passing through the point (0, comma 2), continuing down & right, passing through the x axis between 0 & 1, label on left, x = negative 3, point of inflection, label by max, x = negative 1, local maximum.

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.