5. (This problem represents a very particular case of an interesting area of mathematical ap- plications.) Suppose that you have two points (a, A) and (b, B) specified in the zy-plane. We want to look at the values of fo F(x, y, v) dx for various curves y = y(x) which are dif- ferentiable (here we are writing for the 'velocity' y' = dy/dx) everywhere in the interval. For example, one such scenario might be computing fo2my√√1+ v² dx which is the surface area of a surface of revolution. If you want to find the smallest or largest values that the integral can take, it turns out that the function y you are looking for is a function satisfying the partial differential equation OF Dy - d OF = 0. dx Ov Starting from this formula, show that the shortest (differentiable) path connecting two points is a straight line. That is, apply the partial differential equation to F(x, y, v) = √√1+ v², find a differential equation that must satisfied, and conclude (using only Calc I) that v(x) is a constant function. (Come to office hours for help!)

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