Find the maximum

Algebra Level 4

x x , y y and z z are non-negative real numbers so that x + y + z = 3 x+y+z=3 . What is the maximum value of x 2 y 2 z x^{2} y^{2} z ?

Give your answer to 2 decimal places.


The answer is 1.24.

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Freddie Hand by Freddie Hand , 18, United Kingdom

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1 solution

Freddie Hand
Feb 26, 2017

By AM-GM, ( x 2 + x 2 + y 2 + y 2 + z 5 ) 5 ≥ x 2 y 2 z 16 \large (\frac{\frac{x}{2}+\frac{x}{2}+\frac{y}{2}+\frac{y}{2}+z}{5})^{5} \geq \frac{x^{2} y^{2} z}{16} , with equality when x 2 = y 2 = z \frac{x}{2}=\frac{y}{2}=z

Therefore, the maximum value of x 2 y 2 z x^{2} y^{2} z is ( 3 5 ) 5 × 16 = 1.24 (\frac{3}{5})^{5}\times 16=1.24 (2 d.p)

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