Finding minimum value of multi-variable expression without partial derivatives

Question

Minimize where $a$ and $b$ are positive real numbers

$\sqrt{a^{2}\; +\; 4}\; +\; \sqrt{\left( 3-a \right)^{2}\; +\; \left( b-2 \right)^{2}}\; +\; \sqrt{25\; +\; \left( 6-b \right)^{2}}$

I could take the partial derivatives, equate them to 0, and solve the system of equations, but that leads to some messy equations. Is there a more elegant way to do this?

Answer 1

Hint: What is the distance from $(0,0)$ to $(a,2)$ to $(3,b)$ to $(8,6)$?

Answer 2

Hint: use Minkowski/ Triangle Inequality to get a minimum of $10$.