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

See Also

1957 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 35
Followed by
Problem 37
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