Correct solutions were submitted by Lisa Schechner, Bill Webb, Paul Botham.
The only functions satisfying the given conditions are the
quadratic functions. The argument requires the function to be
differentiable only once.
We start with the given equation, using x in place of
a:
.
A little algebra gives
.
Dividing by (b - x)3 then
gives
.
The key is to recognize that the left hand side of this equation is
and so integrating both sides of this gives
with
C a constant of integration. Multiply both sides by (b - x)2 and
conclude that f must be a quadratic function. It's
straightforward to verify that every quadratic function satisfies the given
equation.
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©2005 Alberto L. Delgado