Given-x-y-z-R-such-that-x-2-3xy-4y-2-z-0-when-xy-z-reaches-its-max-value-find-the-max-value-of-2-x-1-y-2-z-

Question Number 180728 by CrispyXYZ last updated on 16/Nov/22

Given x, y, z \in R^{+} such that x^{2} - 3 x y + 4 y^{2} - z = 0 .

when \frac{x y}{z} reaches its \max value, find the \max value of

\frac{2}{x} + \frac{1}{y} - \frac{2}{z} .

Answered by mr W last updated on 20/Nov/22

\frac{x y}{z} = m a x m e a n s \frac{z}{x y} = m i n

\frac{z}{x y} = \frac{x^{2} - 3 x y + 4 y^{2}}{x y} = \frac{x}{y} + \frac{4 y}{x} - 3 ⩾ 2 \sqrt{\frac{x}{y} \times \frac{4 y}{x}} - 3 = 1

m i n = 1 a t \frac{x}{y} = \frac{4 y}{x}, i . e . \frac{x}{y} = 2

\frac{2}{x} + \frac{1}{y} - \frac{2}{z} = \frac{2}{x} + \frac{1}{y} - \frac{2}{x^{2} - 3 x y + 4 y^{2}}

= \frac{2}{2 y} + \frac{1}{y} - \frac{2}{4 y^{2} - 6 y^{2} + 4 y^{2}} = \frac{2}{y} - \frac{1}{y^{2}}

= - (\frac{1}{y^{2}} - \frac{2}{y} + 1) + 1 = 1 - {(\frac{1}{y} - 1)}^{2} ⩽ 1

i . e . {(\frac{2}{x} + \frac{1}{y} - \frac{2}{z})}_{m a x} = 1

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