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Second Derivatives and Beyond

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 "' 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.
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