Divide the number 30 into two parts such that their product is ma… - Vidyadip

Question

Divide the number $30$ into two parts such that their product is maximum.

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Answer

Let the first part of $30-x$.
$\therefore$ Their product
$ =x(30- x )$
$=30 x - x ^2$
$= f ^{\prime}( x ) \quad \ldots . . \text { (Say) }$
$\therefore f ^{\prime}( x )=\frac{ d }{ d x}(30-2 x)$
$=0-2 \times 1$
$=-2 $
The root of the equation $f^{\prime}(x)=0$
i.e. $30-2 x=0$ is $x=15$
and $f^{\prime}(15)=-2<0$
$\therefore$ By the second derivative test, $f$ is maximum at $x =15$.
Hence, the required parts of $30$ are $15$ and $15.$

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