<s>[INST] Find the smallest value of the following expression $\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2}+ \sqrt{(y-3)^2 +9}$ 
Find the smallest value of the following expression:
  $$\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2} + \sqrt{(y-3)^2 +9}$$

I tried to derive the expression with respect to $x$ and $y$ and then equal the derivative to zero and find the critical points, but I could not do that. It is complicated. Is there a way?
 [/INST] Use the Minkowski inequality:
$$
\begin{aligned}
&\sqrt{(x-9)^{2}+4}+\sqrt{x^{2}+y^{2}}+\sqrt{(y-3)^{2}+9}\\
=&\sqrt{(9-x)^{2}+2^2}+\sqrt{x^{2}+y^{2}}+\sqrt{3^2+(3-y)^{2}}\\
\geqslant&\sqrt{(9-x+x)^2+(2+y)^2}+\sqrt{3^2+(3-y)^{2}}\\
\geqslant&\sqrt{(9-x+x+3)^2+(2+y+3-y)^2}=13
\end{aligned}
$$
"$=$" holds iff $(9-x,2)$ and $(x,y)$ are linearly dependent and $(9,2+y)$ and $(3,3-y)$ are linearly dependent, i.e. $x=\dfrac{21}{5},y=\dfrac{7}{4}$.
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