Second Derivatives and Beyond
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.