Find $y(x)$ such that the Euclidean distance between $(x_a, y(x_a))$ and $(x_b,y(x_b))$ is a minimum, i.e., find the minimum of
$$J = \int_{x_a}^{x_b} \left[1 + \left(\frac{dy}{dx}\right)^2\right]^\frac{1}{2}dx$$
with respect to $y(x)$.
So far I have attempted to integrate this. Is the first integral all I need?
Assume wlog that $x_b>x_a$ and let $$d=\sqrt{(y(x_b)-y(x_a))^2+(x_b-x_a)^2}$$ denote the Euclidean distance between the two points $(x_b,y(x_b))$ and $(x_a,y(x_a))$. It is easy to see that $$d\geq|x_b-x_a|=x_b-x_a=\int^{x_b}_{x_a}dx$$ On the other hand $$d\leq\int^{x_b}_{x_a}\Big(1+\Big(\frac{dy}{dx}\Big)^2\Big)^{1/2}\,dx$$ The last inequality follows from the fact that the length of the straight line joining two points in space is no longer than any other curve joining these two points. Therefore from the last two inequlaties we get that the minimum is reached when $y(x)=c$ for some constant $c$.
