The sum of two positive numbers is 12. The numbers

given $\mathrm{x}+\mathrm{y}=12$

$$\begin{gathered} \Rightarrow \mathrm{y}=12-\mathrm{x} \\ \mathrm{S}={\mathrm{x}}^2+{\mathrm{y}}^2 \\ ={\mathrm{x}}^2+{(12-\mathrm{x})}^2 \\ {\mathrm{S}}^1=2\mathrm{x}+2(12-\mathrm{x})(-1) \end{gathered}$$

S is min or max

$$\begin{gathered} \Rightarrow {\mathrm{S}}^1=0 \\ 2\mathrm{x}-24+2\mathrm{x}=0 \\ 4\mathrm{x}=24 \\ \mathrm{x}=6 \\ \mathrm{x}=6 \mathrm{y}=12-\mathrm{x} \\ =12-6=6 \end{gathered}$$