Continuous Calculus

Calculus Level 4

True or False?

Suppose $f$ is a real-valued continuous function defined on $[0, 1]$ such that $(2x - 1)(2f(x) - x) > 0$ for all $x \neq \frac 12.$

If the first derivative $f'\left(\frac 12 \right)$ exists, then $f'\left(\frac 12 \right)$ cannot be smaller than $\frac 12$ .