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.

Let $X(a,b),O(0,0),A(1,0),B(3,4)$.

Then, as you pointed out, we want to minimize $XO+XA+XA+XB$.

• For any $X$, it holds that $XO+XA\geqslant OA$ whose equality is attained when $X$ is on the line segment $OA$.

• For any $X$, it holds that $XA+XB\geqslant AB$ whose equality is attained when $X$ is on the line segment $AB$.

Therefore, for any $X$, it holds that $$XO+XA+XA+XB\geqslant OA+AB=1+2\sqrt 5$$ whose equality is attained when $X=A$.

Let me give an graphical representation... I mean this is more like an physics-based informal proof. It shows that if you pull the rope out the ring is gonna go down to the lowest point and the rope will align with the dotted lines. Also you can think of it geometrically: as the former answer said, the length of any side of a triangle is always less than the sum of the other two, so there we goes.