Question

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

Answer 1

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