Constrained optimization max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$ max  $ f(x,y) = x+y$    subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$
I need to solve this by the Kuhn Tucker conditions without  using concavity of the Lagrangian.

The constraint $x^2 + y^2 \leq 4$ may be replaced by $x^2 + y^2 = 4$ since if $f$ were maximized when $x^2 + y^2 < 4$ we could simply increase $x$ or $y$ so that $x^2 + y^2 = 4$, contradicting the fact that $x+y$ was maximized.
As such we may write $y = \sqrt{4-x^2}$. Now we need to maximize $f = x + (4-x^2)^{1/2}$. The maximum occurs when $f'(x)=0$, that is when:
$$f'(x) = 1 + -x(4-x^2)^{-1/2} = 0 $$
This is equivalent to the fact that:
$$x = \sqrt{4-x^2}$$
Which after simplifying gives $2x^2 = 4$ and so $ x = \sqrt{2}$. As the equations are symmetric $y = \sqrt{2}$ also. 
