Problem:
Draw the graphs of the function

and its second derivative.

1. Graphically verify that the function is concave upward if and only if its second derivative is positive.
2. Graphically verify that the function is concave upward if and only if its second derivative is positive.
3. Find the inflection points.

Visualization:
Use the following LiveMath notebook to do the verification. The graph of the original function is blue and the graph of its second derivative is red. f has one inflection point; can you find it?

It is possible for you to change the function. With the mouse highlight the part of the function which you want to change and type in the new terms. You may get an error message but just click with the mouse on the graph. Some examples which you may try:

• x4-3x2+3
[Type in: x^4-3x^2+3 Note that you will need to position the cursor after you type in a power.]

• e-x2
[Type in: e(-x^2)]

• x1/3
[Type in: x^(1/3)]

• |x2-4|
[Type in: x^2-4, highlight what you just typed in and then type in|.]