CPM Homework Help : CALC Problem 3-113

Use your observations from problem 3-98 to algebraically verify that $y = x ^ { 3 } + \frac { 3 } { 2 } x ^ { 2 } - 6 x + 2$ is concave up when $x = 0$ .

Hint 1:

Your study team probably observed that concavity has to do with the way that slopes change.

Hint 2:

If $f^\prime(x)$ increases, then the graph of $f(x)$ is curving up and will be concave up.
If $f^\prime(x)$ decreases, then the graph of $f(x)$ is curving down and will be concave down.

Step 1:

To determine if $f(x)$ is concave up at $x = 0$ , you can test the slope at points before and after $x = 0$ and see if those slopes are increasing or decreasing.

Step 2:

Find $f^\prime(x)$ and choose points close to $x = 0$ .
Evaluate those points. For example, $f^\prime(−0.1) =$ ___________ and $f^\prime(0.1) =$ __________
Are they increasing or decreasing?

Answer:

If $f^\prime(−0.1) < f^\prime(0.1)$ then the slopes of $f(x)$ are curving down and the $f(x)$ is concave up at $x = 0$ .
If $f^\prime(−0.1) > f^\prime(0.1)$ then the slopes of $f(x)$ are curving down and the $f(x)$ is concave down at $x = 0$ .