Show that among all positive numbers x and y with x^2 + y^2 = r^2, the sum x + y is largest when x = y = r/√2. ​

Question

Show that among all positive numbers x and y with x² + y² = r², the sum x + y is largest when x = y = \(\frac{r}{\sqrt2}\).

Answer 1

x, y are positive numbers 

x² + y² = r² is the given condition 

Therefore,

y = \(\sqrt{r^2-x^2}\) 

(Note that the - \(\sqrt{r^2-x^2}\) is not considered as the x and y are + ve)

Let