CPM Homework Help : APCALC Problem 5-98

Using the graph of $y=f^\prime(x)$ below, determine the values of $x$ where $f$ has a local minimum, local maximum, or point of inflection. Justify your answer for each point.

Downward parabola, labeled f prime of x, vertex at the point (negative 1, comma 4), passing through the points (0, comma 3), (negative 3, comma 0), & (1, comma 0).

Hint 1:

You are looking at a graph of $f^\prime\left(x\right)$ , which shows all the slopes of $f\left(x\right)$ .

Hint 2:

A local minimum on $f\left(x\right)$ is located where the slopes of $f\left(x\right)$ change from negative to positive.

Hint 3:

A local maximum on $f\left(x\right)$ is located where the slopes of $f\left(x\right)$ change from positive to negative.

Hint 4:

An inflection point on $f\left(x\right)$ is where concavity changes, and concavity is a determined by the slope of the the slopes.

Answer:

$x = −3$ : local minimum
$x = 1$ : local maximum
$x = −1$ : inflection point