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/(sqrt(2))` .

Explanation

Here, `x^2+y^2 = r^2`

`=> y = sqrt(r^2-x^2)`

Let `S = x+y`

`=> S = x+sqrt(r^2-x^2)`

For `S` to be largest, `(dS)/dx =0`.

`:. (dS)/dx = 1+1/sqrt(r^2-x^2)(-2x)`

`:. 1+1/(2sqrt(r^2-x^2))(-2x) = 0`

`=>x = sqrt(r^2-x^2)`

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