Application of Derivatives — MATHS STD 12 Science — Question
Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSApplication of Derivatives3 Marks
Question
Find two numbers whose sum is $24$ and whose product is as large as possible.
✓
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 $12$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.