Problem 14 (2007 AIME II)

Let $f$ be a non-constant polynomial with real coefficients such that $f(x) \cdot f(-x) = f(x^2)$ for all real $x$, and $f\!\left(\tfrac{1}{2}\right) = \tfrac{1}{4}$. Find $f\!\left(\tfrac{1}{4}\right)$.