Concavity

The slope or derivative of a function f describes whether f is increasing, decreasing, or constant. The concavity of a function f describes whether f is curving up, curving down, or not curving at all. Consider our morning bowl of fruit loops. We're lucky that the cereal bowl inventor of the cereal bowl made it concave up. If it had no concavity, it would be a plate. The world would be chaos. Worse, if the bowl was concave down we would eat our loops off of the floor...maybe David Bowie could handle it. The concavity of a bowl is determined by the amount of cereal contained. The concavity of a function corresponds to the sign of its second derivative.

Be Careful: Increasing and positive don't mean the same thing. We can have functions that are increasing but negative:

Similarly, decreasing and negative don't mean the same thing. We can have functions that are decreasing but positive: