Find two numbers whose sum is 24 and whose product is as large as… - Vidyadip

Question

Find two numbers whose sum is $24$ and whose product is as large as possible.

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Answer

Let the two numbers be $x$ and $y$. According to the question, $x + y = 24$
$\Rightarrow y = 24 - x …(i)$
And let $z$ is the product of $x$ and $y.$
$\Rightarrow z = xy$
$\Rightarrow z = x(24 - x) [$From eq. $(i)]$
$\Rightarrow z = 24x - x^2$
$\Rightarrow \frac{{dz}}{{dx}} = 24 - 2x$ and $\frac{{{d^2}z}}{{d{x^2}}} = - 2$
Now to find turning point, $\frac{{dz}}{{dx}} = 0$
$\Rightarrow 24 - 2x = 0 \Rightarrow x = 12$
At $x = 12,\frac{{{d^2}z}}{{d{x^2}}} = - 2$ [Negative]
$\therefore x = 12$ is a point of local maxima and z is maximum at $x = 12.$
$\therefore $ From eq. $(i), y = 24 - 12 = 12$
Therefore, the two required numbers are $12$ and $12.$

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