$f(x)$ is a monic real polynomial with $f(1/x) = f''(x) = c$, when $x$ tends to infinity, and $f(x)\ge f(1)$ for all real $x$. What is $f$?
My attempt: I showed that, as $x$ tends to infinity $f(1/x) = c = f''(x)$, f has degree $\le 3$ and constant $c$. Thus let $f(x) = x^3 + ax^2 + bx + c$. Then using the last given condition I tried to find the co-efficients, but that seemed to be a long process. I wonder if there's a shorter and more elegant version of this proof.