A correct solution came from Bradley University alumnus Nathan Pauli. Other correct solutions were submitted by Elie Ghosn, Dan Dima, Siupor Lam, Steven Prowse, Juan Carlos Marivela, Ahron Teitelman, Steven Young, Lou Cairoli, Keith Anker, Nancy Scharzkopf and one unsigned submission from Uruguay.
Look at the polynomial g(x) = f (x) - f '(x). This is a cubic polynomial which (up to a switch in sign) goes to +∞ as x gets arbitrarily large and goes to -∞ as x gets arbitrarily small (negative). By the Intermediate Value Theorem, it crosses the x-axis at some point. This root of g(x) is a point where f (x) agrees with its derivative.
A more difficult question is the following: Suppose that f (x) is a polynomial with a real root. Show that f (x) agrees with its derivative at some point.
©2005 Alberto L. Delgado