Second Derivatives and Beyond
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.