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# Find two positive numbers $x$ and $y$ such that their sum is 35 and the product $x^2 y^5$ is a maximum.

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Toolbox:
• $\large\frac{d}{dx}$$(x^n)=na^{n-1} Step 1: x+y=35 y=35-x P=x^2y^5 P=x^2(35-x)^5 \large\frac{dP}{dx}$$=x^2.5(35-x)^4(-1)+(35-x)^5.2x$
$\quad\;=x^2(35-x)^4[-5x+2(35-x)]$
$\quad\;=x^2(35-x)^4[70-7x]$
Step 2:
When $x$ is slightly < 10 $\large\frac{dP}{dx}=$$(+)(+)(+)=+ve When x is slightly > 10 \large\frac{dP}{dx}=$$(+)(+)(-)=-ve$
$\Rightarrow \large\frac{dP}{dx}$ changes sign from +ve to -ve as $x$ increases through 10.
$\Rightarrow P$ is maximum at $x=10$
From (1) $y=35-10=25$
Hence the required numbers are 10 & 25.
answered Aug 8, 2013