This applet has been written to explore the following math olympiad problem (sorry, I cannot recall the source for now).
Let p(x) = a x^2 + b x + c be a quadratic polynomial such that -1 <= p(x) <= 1 for 0 <= x <= 1. Prove that |a| + |b| + |c| <= 17.
We create sliders for a, b and c, then define p(x) = a x^2 + b x + c and empirically vary a, b, c so that the graph of p(x) for 0 <= x <= 1 always stays within the green box.
All the while we keep track of the value of |a| + |b| + |c|, attempting to maximize it.