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 = x² + y²
⇒ z = x² + (15 - x)² [from eq.(1)]
⇒ z = x² + x² + 225 - 30x
⇒ z = 2x² + 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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