If x+y=8, then the maximum value of xy is ____________ .

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16 \text { Given }: x + y = 8 \Rightarrow y = 8 - x.........................\left( 1 \right) \text { Let } f\left( x \right) \text { be } xy . \Rightarrow f\left( x \right) = x\left( 8 - x \right).....................\left[ \text { From eq } . \left( 1 \right) \right] \Rightarrow f'\left( x \right) = 8 - 2x \text { For a local maxima or a local minima, we must have } f'\left( x \right) = 0 \Rightarrow 8 - 2x = 0 \Rightarrow 8 = 2x \Rightarrow x = 4 \Rightarrow y = 8 - 4 = 4..................\left[ \text { From eq.} \left( 1 \right) \right] \text { Now,} f''\left( x \right) = - 2 \Rightarrow f''\left( 4 \right) = - 2 < 0 \text { So, x = 4 is a local maxima }. \text { Hence, the local maximumvalue is given by } f\left( 4 \right) = 4 $\times$ 4 = 16

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