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  4. If x+y=10 , then the maximum value of xy is

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Q. If $ x+y=10 $ , then the maximum value of $ xy $ is

Jharkhand CECEJharkhand CECE 2014

Solution:

Given, $ x+y=10 $ $ y=10-x $ ... (i)
Now, $ f(x)=xy=x(10-x)=10x-{{x}^{2}} $
$ \therefore $ $ f'(x)=10-2x $
For maximum value of $ f(x) $ , put $ f'(x)=0 $
$ \Rightarrow $ $ 10-2x=0\Rightarrow x=5 $
$ \therefore $ $ x=5 $ and $ y=5 $ [from Eq. (i)]
Hence, maximum value of $ xy=5\times 5=25 $