Study Guide

Concavity

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:

  • 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 "' 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 ' goes from negative to zero to positive, therefore ' is increasing:

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

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

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

    • f is concave up.
        
    • ' is increasing.
        
    • " is positive.
  • 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 "' is decreasing" or "the slope of f is decreasing." If we have a bowl, then ' goes from positive to zero to negative, so ' is decreasing:

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

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

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

    • f is concave down.
        
    • ' is decreasing.
        
    • " is negative.
  • No Concavity

    If " is positive, then f is concave up. If " is negative, then f is concave down. If " is zero, we say that the function f has no concavity. It's flat. Pancakes can survive in a world of no concavity. When a function has no concavity, it means f doesn't curve at all. It's a straight line: