Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 92 Marks
Question
Find two positive numbers whose sum is $14$ and the sum of whose squares is minimum.
✓
Answer
Let the numbers be $x$ and $y$. Then,
$x + y = 14 ... (i)$
Let $S$ be the sum of the squares of $x$ and $y$.
Then,
$S=x^2+y^2$
$\Rightarrow S = x ^2+(14- x )^2$
$\Rightarrow S=2 x ^2-28 x +196$
$\Rightarrow \frac{d S}{d x}=4 x -28 $ and $\frac{d^2 S}{d x^2}=4$
The critical points of $S$ are given by $\frac{d S}{d x}=0$.
$\because \frac{d S}{d x}=0 $
$\Rightarrow 4 x -28=0$
$ \Rightarrow x =7$
Clearly $\frac{d^2 S}{d x^2}=4 ;0$
Thus $, S$ is minimum when $x = 7$ Putting $x = 7$ in equation $(i),$ we obtain $y = 7$
Hence. the required numbers are both equal to $7$.
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