# Second Derivatives and Beyond

# Concave Up

We say a function *f* is **concave up** if it curves upward like a right-side up spoon:

It's also possible to have only part of the spoon. Both of these functions are concave up:

"*f* is concave up" means exactly the same thing as "*f *' is increasing" or "the slope of *f* is increasing." If we have a bowl that is right-side-up (concave side up), properly holding our fruit loops, then *f *' goes from negative to zero to positive, therefore *f *' is increasing:

If *f* is increasing and concave up, then the slope of *f* becomes steeper - in other words, *f *' is increasing:

If *f* is decreasing and concave up, then the slope of *f* starts negative and approaches zero—in other words, *f *' is increasing:

Saying that a differentiable function is increasing is the same as saying the derivative of that function is positive. Assuming that *f *' is differentiable, saying that *f *' is increasing is the same as saying *f *" is positive. Therefore the following statements all mean the same thing:

*f*is concave up.

*f*' is increasing.

*f*" is positive.