Proving that $\frac{x}{C-x}$ is a convex function (m/m/1 delay)

Question

Consider the function

$$\frac{x}{C-x} \quad (C\in\Bbb R \ \text{a constant}).$$

C>=0

How can I prove that it is a convex function?

or this $$\frac{x}{10-x}$$

Answer 1

Your function is undefined at $x=C$. In all other places, show that $f''(x)>0$. Arithmetic may be easier if you prove $$f(x) = \frac{C}{C-x} - 1$$