<s>[INST] Find the minimum of $\sqrt{4y^2-12y+10}+\sqrt{18x^2-18x+5}+\sqrt{18x^2+4y^2-12xy+6x-4y+1}$ Find the minimum of $$f(x,y)=\sqrt{4y^2-12y+10}+\sqrt{18x^2-18x+5}+\sqrt{18x^2+4y^2-12xy+6x-4y+1}$$ It seems that $f_x=f_y=0$ is very hard to compute. Is there any easier idea?
 [/INST] We have
\begin{align}
f(x, y) &= \sqrt{4y^{2}- 12y+ 10}+ \sqrt{18x^{2}- 18x+ 5} \\ 
&\qquad + \sqrt{18x^{2}+ 4y^{2}- 12xy+ 6x- 4y+ 1} \\[6pt]
&= \sqrt{\frac{ ( 2y- 6 )^{2} + ( 6y- 8 )^{2}}{10}} + \sqrt{\frac{( 6x- 5 )^{2}+ ( 12x- 5 )^{2}}{10}} \\[6pt]
&\qquad + \sqrt{\frac{ ( 6x+ 2y- 1  )^{2}+  ( 12x- 6y+ 3  )^{2}}{10}}\\
&\ge \frac{6- 2y}{\sqrt{10}}+ \frac{5- 6x}{\sqrt{10}}+ \frac{6x+ 2y- 1}{\sqrt{10}} \\[10pt]
&= \sqrt{10}.
\end{align}
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