CPM Homework Help : CALC3RD Problem 5-43

Using the graph of the slope function at right, determine where the following situations occur for $g$ :

Hint:

Notice that this is a graph of $g^\prime\left(x\right)$ , but you are being asked about $y = g\left(x\right)$ .

Relative minima.
Hint (a):
Relative minima and maxima can exist when $g^\prime\left(x\right) = 0$ .
This occurs when $x = a, x = d$ , and $x = f$ . But which points indicate a maximum and which indicate a minimum?
More Help (a):
A minimum occurs when the slope changes from negative to positive.

Relative maxima.
Hint (b):
See the hint in part (a). A maximum occurs when the slope changes from positive to negative.

Interval(s) over which $g$ is increasing.
Hint (c):
$g$ is increasing when $g^\prime(x)>0$ .

Inflection points.
Hint (d):
Inflection points can happen where $g^{\prime \prime}\left(x\right) = 0$ . Remember, we are looking at the graph of $y = g^\prime\left(x\right)$ .

Interval(s) over which $g$ is concave up.
Hint (e):
$g$ is concave up when $g^{\prime \prime}\left(x\right) > 0$ .

Interval(s) over which $g$ is concave down.
Hint (f):
$g$ is concave down when $g^\prime\left(x\right)$ is decreasing.

Coordinate grid, 6 tick marks on the x axis, labeled from left to right, a, b, c, d, e, & f, continuous Curve labeled, g prime of x, coming from lower left, concave down, passing through x axis at a, turning in quadrant 2 at, b, changing concavity in quadrant 1, at, c, passing through x axis at, d, turning in quadrant 4, at e, continuing up & right concave up, passing through x axis at, f.