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# Inequality

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If w, x, y, and z are positive real numbers such that w + 2x + 3y + 4z = 8, then what is the maximum value of wxyz?

Aug 14, 2021

#1
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The minimum value is 9/25.

Aug 14, 2021
#2
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By AM-GM,

$$\frac{w+2x+3y+4z}{4} \ge \sqrt[4]{24wxyz}\\ 2\ge\sqrt[4]{24wxyz}\\ 16\ge24wxyz\\ \frac{2}{3}\ge wxyz$$

so the maximum value is $$\boxed{\frac{2}{3}}$$

for completeness, equality occurs when the terms are equal: $$w=2, x=1, y=\frac{2}{3}, z=\frac{1}{2}$$

Aug 21, 2021