Minimum of this expression I was given a problem to minimise 
$$[(x-y)^2+(12+\sqrt{1-x^2} -\sqrt{4y})^2]$$
Where x,y are real, I have managed to solve it, but it took a lot of time and effort, can anyone provide a short way?
 A: So, the goal here is to minimize: $f(x,y)=(x−y)^2+(12+\sqrt{1−x^2}-\sqrt{4y})^2$.
Notice that $x \in [-1,1]$ and $y \in [0,\infty)$ because of the constraint from both square roots.
We are looking to minimize any positive expressions, $(x−y)^2$ and $\sqrt{1−x^2}$, and maximize and negative expressions, $-\sqrt{4y}$.
Intuitively,
$(x-y)^2$ vs $(12+\sqrt{1−x^2}-\sqrt{4y})^2$ 
While we want to minimize both expressions, the latter expression "grows" more quickly e.g. $[(12+1)^2-12^2] - [(2+1)^2-2^2]>0$
Thus, it is straightforward to check (you can do this by taking the derivative or intuitively) that the minimum occurs at $x=1$. Therefore, we can turn this into an optimization problem of a single variable by substituting the value $x=1$ into the function: $(1-y)^2+(12-\sqrt{4y})^2$
$\dfrac{d}{dy}\left[(1-y)^2+(12-\sqrt{4y})^2\right]=2y-\dfrac{24}{\sqrt{y}}+2=0        \quad\quad$ at $y=\frac{1}{3} [-2+(1945-36 \sqrt{2919})^{1/3}+(1945+36 \sqrt{2919})^{1/3}]\approx 4.5969$
$f(1,4.5969)\approx72.4115$ which is smaller than the previously mentioned $f(0,0)=169$
A: If it's unclear , ask for any clarifications.

