Let $k > 0$ be given. Let $$f(x)=\left(2^\frac{k}{1-x}-1\right)\left(\frac{1-x}{x}\right)$$ where $0 < x < 1$. Simulation results shows that the function is having only one minima. I want to prove $f(x)$ is a convex function mathematically. Here $h(x)=\left(2^\frac{k}{1-x}-1\right)$ is a increasing function and $g(x)=\left(\frac{1-x}{x}\right)$ is a monotonically decreasing function. As a result $g'(x)<0$. These are observed by matlab simulations. But when I am considering these functions together I am failing to prove that as a convex function.