Optimising the given function I am required to find minimum value of the following function without calculus.
$$f(a,b)=\sqrt{a^2+b^2}+2\sqrt{a^2+b^2-2a+1}+\sqrt{a^2+b^2-6a-8b+25}$$
My attempt:
I realised that I can write the function as $$f(a,b)=\sqrt{(a-0)^2+(b-0)^2}+2\sqrt{(a-1)^2+(b-0)^2}+\sqrt{(a-3)^2+(b-4)^2}$$ which when $a$ and $b$ replaced with coordinates in Cartesian plane represents distances from points $(1,0),(0,0)$ and $(3,4).$ But I'm not sure how I can minimise this.

The function to be minimized is composed of three addends, one of which has a double coefficient. If we eliminate it, imposing that
$\sqrt{a^{2}+b^{2}-2a+1}=0$
give zero contribution, we get:
$a=1$ and $b=0$.
Therefore the minimum is:
$f(1,0)=\sqrt{1^2+0^2}+2\sqrt{1^2+0^2-2+1}+\sqrt{1^2+0^2-6-0+25}=1+\sqrt{20}=1+2\sqrt{5}$.
