Problem of the Week

PROBLEM 222

The following problem was posed by Jens Voβ.  Danke, Jens!

True or false:  Every cubic polynomial agrees with its derivative at some point; that is, if f (x) is a cubic polynomial function, there exists a real number c so that  f (c) = f ' (c). 

If it's true, give an explanation; if it's false, a counterexample.

(For the polynomially powerful: Same question for a polynomial of arbitrary odd degree.)

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ã2005 Alberto L. Delgado