1957 AHSME Problems/Problem 36

By AM-GM, we have $\frac{x+y}{2} \geq \sqrt{xy}$ Substituting, we have $\frac{1}{2} \geq \sqrt {xy}$ $\frac{1}{4} \geq xy$ Equality occurs when $x = y = \frac{1}{2}$ $\boxed{D}$