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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.

1 Answer

  • $\large\frac{d}{dx}$$(x^n)=na^{n-1}$
Step 1:
Step 2:
When $x$ is slightly < 10 $\large\frac{dP}{dx}=$$(+)(+)(+)=+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 by sreemathi.v

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