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.



Functional minimization problem
Actually this is one of the oldest problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optimal profile for the motion of bodies (projectiles, ships, etc), that is, the profile giving the minimal aerodynamic or hydrodynamic resistance. Here you are assuming axial symmetry, but the problem is even studied under more general assumptions.


So you can find a lot of material from the keywords "Newton problem" and "Calculus of variations". Here is a nice [survey paper](http://cvgmt.sns.it/media/doc/paper/1339/BUTTAZZO.PDF) by Giuseppe Buttazzo (see in particular section *2* and the references therein for the radial case).


