# Second Derivatives and Beyond

# Concave Down

We say a function *f* is **concave down** if it curves downward like an upside-down spoon (concave side down):

It's also fine to have only part of the bowl. Both of these functions are concave down:

"*f* is concave down" means exactly the same thing as "*f *' is decreasing" or "the slope of *f* is decreasing." If we have a bowl, then *f *' goes from positive to zero to negative, so *f *' is decreasing:

If *f* is increasing and concave down, then the slope of *f* starts positive and decreases—in other words, *f *' is decreasing:

If *f* is decreasing and concave down, then the slope of *f* starts negative and becomes steeper (more negative)—in other words, *f *' is decreasing:

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

*f*is concave down.

*f*' is decreasing.

*f*" is negative.