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.

✓

Answer

Let the first part of 20 be $x$.
Then the second part is $20-x$.
$\therefore$ their product $= x (20- x )=20 x - x ^2= f ( x )$
$
\begin{aligned}
& \therefore f ^{\prime}( x )=\frac{d}{d x}\left(20 x - x ^2\right) \\
& =20 \times 1-2 x \\
& =20-2 x \\
& \text { and } f ^{\prime \prime}( x )=\frac{d}{d x}(20-2 x ) \\
& =0-2 \times 1 \\
& =-2
\end{aligned}
$
The root of the equation $f^{\prime}(x)=0$
i.e. $20-2 x=0$ is $x=10$
and $f^{\prime \prime}(10)=-2<0$
$\therefore$ by the second derivative test, $f$ is maximum at $x =10$.
Hence, the required parts of 20 are 10 and 10 .

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