1957 AHSME Problems/Problem 36

Problem

If $x + y = 1$ , then the largest value of $xy$ is:

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0$

Solution

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 $x = y = \boxed{\textbf{(D) }\frac12}$ .

~JustinLee2017

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 35	Followed by Problem 37
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1957 AHSME Problems/Problem 36

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