<s>[INST] If $X+Y = 10 $ ($X$ and $Y $ both are positive) then what is the maximum value of $(X^3)(Y^2)$? I can get to the result by trying different values of X and Y but that is of course time taking. I want to know if there is a better way to get to the result?
 [/INST] $Y = 10 - X$,and $X^3Y^2 = X^3(10-X)^2 = X^3(X^2 - 20X + 100) = X^5 - 20X^4 + 100X^3 = f(X)$. 
$f'(X) = 5X^4 - 80X^3 + 300X^2 = 0 \iff 5X^2(X^2 - 16X + 60) = 5X^2(X-6)(X-10) = 0 \iff X = 0, 6, 10$. Since $0 < X < 10$, $X = 6$, and $Y = 10 - 6 = 4$. From this the max value of $X^3Y^2$ is $6^34^2 = 3,456$.
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