Euler Equation to Compute Extreme?

[black_hole]

A problem on my homework:

We learn early on that "the shortest distance between two points is a straight line." Let's prove it...Using the Euler equation, compute the extrema of

∫sqrt(1 + (dy/dx)²)dx from x₁ to x₂ ...show that this corresponds to lines "y = mx +b".

Euler had a lot of different equations. I recognize this as the formula for calculating the length of a graph but that's not what I'm being asked to do. Does anyone what formula this is referring to?

Thanks!

 

[Hurkyl]

If the hint is confusing, then ignore it for the moment. Amongst all paths from one point to another, how would you go about determining which ones have the shortest length?

 

[HallsofIvy]

I would presume that they are referring to the "Euler equation" (also known as the "Euler-Lagrange equation") for "Calculus of variations" since that is the kind of problem Calculus of variations solves.

http://mathworld.wolfram.com/CalculusofVariations.html

 

[black_hole]

Thanks HallsofIvy. That was exactly what I was looking for!