PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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If X+Y=8, Then the Maximum Value of Xy is (A) 8 (B) 16 (C) 20 (D) 24 - PUC Karnataka Science Class 12 - Mathematics

Question

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

• 8

• 16

• 20

• 24

Solution

$16$

$\text { Given }: x + y = 8$
$\Rightarrow y = 8 - x.........................\left( 1 \right)$
$\text { Let } f\left( x \right) \text { be } xy .$
$\Rightarrow f\left( x \right) = x\left( 8 - x \right).....................\left[ \text { From eq } . \left( 1 \right) \right]$
$\Rightarrow f'\left( x \right) = 8 - 2x$
$\text { For a local maxima or a local minima, we must have }$
$f'\left( x \right) = 0$
$\Rightarrow 8 - 2x = 0$
$\Rightarrow 8 = 2x$
$\Rightarrow x = 4$
$\Rightarrow y = 8 - 4 = 4..................\left[ \text { From eq.} \left( 1 \right) \right]$
$\text { Now,}$
$f''\left( x \right) = - 2$
$\Rightarrow f''\left( 4 \right) = - 2 < 0$
$\text { So, x = 4 is a local maxima }.$
$\text { Hence, the local maximumvalue is given by }$
$f\left( 4 \right) = 4 \times 4 = 16$
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Solution If X+Y=8, Then the Maximum Value of Xy is (A) 8 (B) 16 (C) 20 (D) 24 Concept: Graph of Maxima and Minima.
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