  ### Home > APCALC > Chapter 3 > Lesson 3.3.3 > Problem3-113

3-113.

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

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

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.

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.

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?

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 $x = 0$.