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 converted 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.

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.

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 but would love to see an analytic solution (algebraic or not, doesn't matter). Thank you so much in advance. Fingers crossed.

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 converted 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.