1+(f′(x))^2−f′′(x)=0 Make the substitution u=f′ in equation (B). Solve the resulting first-order equation for u(x).
Integrate your answer for problem 1 to find f(x) solving the original Euler-Lagrange equation.
What is the cheapest path from (−1,0) to (1,0)?
What happens when you try to use the formula you found in problem 2 to write out the cheapest path from (−3,0) to (3,0)?
Explain why the formula doesn't work in problem 4 in terms of the original problem of finding a cheapest path.