How to show that the following function is strictly convex within a range?

Question

Given a function $f(x)=\frac{x+1}{2x(11-x)}$ on range $x\in[1,10]$, how do I show that it is strictly convex? I suspect it based on plotting.

Answer 1

You get the second derivative $f''(x)=\frac{1}{(11-x)^3}+\frac{(11-2x)^2}{(x(11-x))^3}+\frac{1}{(x(11-x))^2}$, which is greater than zero when $x\in[1,10]$. This implies the $f(x)$ is strictly convex when $x\in[1,10]$.