# Functional minimization problem

Is there a smooth solution to minimize this: $$\int_0^1{x \over {1+k^2f'(x)^2}}dx, f(0)=1, f(1)=0, f'(x)\leq 0, k^2>0.$$ I could "solve" it using a numeric approximation (my algorithm converged so apparently there is a valid local minimum in the function space) but would love to see an analytic solution (algebraic or not, doesn't matter). Thank you so much in advance. Fingers crossed.

• I can't tell what the problem is. If you let $k$ get arbitrarily large, your integrals tend to zero. So there would appear to be no minimum. – Ryan Budney Apr 21 '15 at 6:28
• @RyanBudney: Maybe $k$ is supposed to be fixed? – Nate Eldredge Apr 21 '15 at 6:34
• $k^2$ is a constant, a form factor. – Mandrill Apr 21 '15 at 6:34
• Have you tried Euler-La grange equation? – Fan Zheng Apr 21 '15 at 6:40
• @ Fan Zheng I did, but it doesn't work (at least not the standard way to use it) because a "zigzag" (going up and down with a small x variation) function would make a very high $f'(x)$ and that would minimize the functional. I don't know how to introduce a monotonic restriction in the Euler-Lagrange method. – Mandrill Apr 21 '15 at 6:46