Divide the number 20 into two parts such that their product is ma… - Vidyadip

Question

Divide the number $20$ into two parts such that their product is maximum

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Answer

The given number is 20.Let x be one part of the number and y be the other part.
$\therefore x+y=20$
$\therefore y=(20-x).$
The product of two numbers is $x y$.
$\therefore f(x)=x y$
$=x(20-x) \ldots \ldots .[\text { From (i)] }$
$=20 x-x^2$
$\therefore f^{\prime}(x)=20-2 x \text { and } f^{\prime \prime}(x)=-2$
Consider, $f ^{\prime}( x )=0$
$\therefore 20-2 x=0$
$\therefore 20=2 x$
$\therefore x=10$
For $x=10$,
$f^{\prime \prime}(10)=-2<0$
$\therefore f ( x )$, i.e., product is maximum at $x =10$
and $y=20-10 \ldots \ldots$. [From (i)]
i.e., $y=10$

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