<s>[INST] Maximize linear function over disk of radius $2$ 
Maximize $f(x, y) = x + 2y$ with constraint $x^2 + y^2 \le 4$.

$f(x, y)$ has no CP's so thats something gone.
I considered the boundary/edge of $g(x, y) = x^2 + y^2$. And I got the point $(0, 2), (2, 0)$, which gave $f(0, 2) = f(2, 0) = 4$. However, the point $(2/\sqrt(5), 4/\sqrt(5))$ gives a larger max.
How do I approach this to get this point?
 [/INST] 
Max $z=x+2y$ subject to $x^2+y^2\le4$.

The contour lines: $y=-\frac{1}{2}x+\frac{1}{2}z$, the largest value of $z$ occurs when $y$ is tangent to the constraint circle $x^2+y^2=4$ or $y=\sqrt{4-x^2}$.
Thus the slope of tangent line must be equal to the slope of $y=-\frac{1}{2}x+\frac12z:$
$$y'=\frac{-2x}{2\sqrt{4-x^2}}=-\frac12 \Rightarrow x=\frac{2}{\sqrt{5}} \Rightarrow y=\frac{4}{\sqrt{5}}.$$
Graphical illustration:

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