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Second Derivatives and Beyond

Second Derivatives and Beyond

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.

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