Determine two positive numbers whose sum is 15 and the sum of who… - Vidyadip

Question

Determine two positive numbers whose sum is $15$ and the sum of whose squares is maximum.

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Answer

Let the two positive numbers be $x$ and $y$. Then,
$x + y = 15 ....(1)$
Now, $z = x^2 + y^2$
$\Rightarrow z = x^2 + (15 - x)^2 [$from eq.$(1)]$
$\Rightarrow z = x^2 + x^2 + 225 - 30x$
$\Rightarrow z = 2x^2 + 225 - 30x$
$\Rightarrow \frac{\text{dz}}{\text{dx}}=4\text{x}-30$
For maximum or minimum value of $z$, We must have
$\Rightarrow \frac{\text{dz}}{\text{dx}}=0$
$\Rightarrow 4\text{x}-30=0$
$\Rightarrow \text{x}=\frac{15}{2}$
$\frac{\text{d}^{2}\text{z}}{\text{dx}^{2}}=4>0$
Substituting $\text{x}=\frac{15}{2}$ in $(1),$ We get
$ \text{y}=\frac{15}{2}$
Thus, $z$ is minimum when $\text{x}=\text{y}=\frac{15}{2}$.

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