How to prove a property of a concave function?

Question

Can you suggest how the inequality $f(\frac{x}{2}) \geq \frac{f(x)}{2}$ for concave increasing $f$ and positve $x$ may be shown?

Answer 1

If $\lambda_1$ and $\lambda_2$ are two arbitrary nonnegative real numbers such that $\lambda_1+\lambda_2$ = 1 then concavity of $f(x)$ implies : $$f(\lambda_1 x+\lambda_2 y)\geq \lambda_1f(x)+\lambda_2f(y)$$

If $f(0)=0$ put $\lambda_1=\lambda_1=0.5$ and $y=0$ you have your result .

Answer 2

It's not true. Try $f(x) = x-1$. Maybe you want to assume $f(0) \ge 0$?