## Concave Up and Down Functions, and Inflection Points

And once again, we can visualize that on the second derivative, the derivative of the derivative. And we see that that is indeed the case. It's reasonably negative right there. Now let's look at concavity from a slightly different perspective. If you think you might too, try to come up with a way to connect these concepts easily.

## Math Insight

So let's review how we can identify concave downward intervals and concave upwards intervals. To appreciate this test, it is first necessary to understand the concept of concavity. Click here to see how to enable Java on your web browser. Then it becomes less positive. Examine the fourteen examples provided in the scroll bar on the top of the applet or enter your own function in the space below the graphs and press new function to display the graphs.

Concave upwards. Given the above, we can decide whether a function is increasing or decreasing by looking at the sign of its derivative.

The Second Derivative Test: We will only know that it is an inflection point once we determine the concavity on both sides of it. The following fact relates the second derivative of a function to its concavity.

This point is called a point of inflection POI. And we see that the derivative is positive as we exit that point. Let me make this clear. Find the first two derivatives: So this is the derivative of this, of the first derivative right over there. Derivatives can be used to find a these.

So, we can see that we have to be careful if we fall into the third case.

So f prime of a is equal to 0. If the derivative is decreasing, that means that the second derivative, the derivative of the derivative, is negative. Obviously if we have the graph in front of us it's not hard for a human brain to identify this as a local maximum point.