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

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

• $XO+XA\geqslant OA=1$ whose equality is attained when $X$ is on the line segment $OA$.

• $XA+XB\geqslant AB=2\sqrt 5$ whose equality is attained when $X$ is on the line segment $AB$.

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

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 $X=A$.