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Local Maxima

Find the maxi...

Question

Find the maximum value of $x^{m} y^{m} w h e r e m, n > 0$ such that x+y =a

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Solution

$x + y = a ⟹ m (\frac{x}{m}) + n (\frac{y}{n}) = a$
$A M \geq G M ⟹ \frac{m (\frac{x}{m}) + n (\frac{y}{n})}{m + n} \geq \sqrt[m + n]{(\frac{x}{m})^{m} (\frac{y}{n})^{n}}$
$x^{m} y^{n} \leq \frac{a^{m + n} . m^{m} . n^{n}}{(m + n)^{m + n}}$
The maximum value is $\frac{a^{m + n} . m^{m} . n^{n}}{(m + n)^{m + n}}$

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