An inflectionless point!
The following example and proof are from David Bressoud.
The following is a function where the derivative is 0 at x = 0, where the derivative is negative for all other values of x, yet does not have a point of inflection at x = 0.
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We note that the derivative is given by:
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We note that 
and 
for 
Before proving that there is no inflection point at zero, allow me to present boffo pictures of f and its derivative
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Notice that f ' is not an increasing function on any interval with 0 as a right-hand endpoint, and it is not decreasing on any interval with 0 as a left-hand endpoint. (More proof will be added tomorrow!)
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