# Second Derivatives and Beyond

### Topics

We talked earlier about how we can tell from the derivative whether the original function is increasing or decreasing. We also talked about concavity), and how if we toss the second derivative into the mi*x*, we can also tell what shape the original function should have.

We need to go over the shapes again here, because the more fluent you are at translating between signs of derivatives and shapes of graphs, the easier this whole business will be.

There are four possible shapes a piece of graph can have. Actually, this is a lie - there are seven possible shapes, but three of them don't need calculus.

Here are the ones that do need calculus:

• *increasing and concave up*

If *f* is increasing and looks like part of a right-side up bowl, it looks like this:

• *increasing and concave down*

If *f* is increasing and looks like part of a upside-down bowl, it looks like this:

• *decreasing and concave up*

If *f* is decreasing and looks like part of a right-side up bowl, it looks like this:

• *decreasing and concave down*

If *f* is decreasing and looks like part of a upside-down bowl, it looks like this:

The shapes that don't need calculus are the ones where *f* has no concavity, but is a straight line. If *f* is a line, it can have three possible shapes:

• *increasing:*

• *decreasing:*

• *constant:*