I have the following problem, and I don't know where to begin.
Consider the curve connecting $(x_0,y_0) = (0,0)$ and $(x_1,y_1) = (1,1)$. Show by explicit computation that the function $y(x) =x$ produces a minimum path length between these points by using the varied function in the form $y(x,\epsilon) =x+δy(x,\epsilon)$, where $δy(x,\epsilon) =\epsilon \sin[π(1−x)]$.
I know how to show that $y(x) = x$ satisfies the requirement for minimum path length by using $F_y -\frac{d}{dx}F_{y'} = 0$, but how do I do this using the varied function? Do I plug in $\frac{dy(x, \epsilon)}{dx}$ into $\int_{0}^{1} \sqrt{1 + (\frac{dy}{dx})^2}dx$ and show that it is greater than $\sqrt{2}$? I am given a hint that I can use taylor expansion, but I mostly just need help setting up the problem. Thanks in advance.