Derivative Value from Average Rates and Concavity

Let $p$ be a twice-differentiable function on the interval $[1, 7]$ with $p(1) = 4$, $p(4) = 10$, and $p(7) = 22$. The average rate of change of $p$ on the interval $[1, 4]$ is 2, and on the interval $[4, 7]$ is 4. If $p''(x) > 0$ for all $x$ in the interval $(1, 7)$, which of the following statements must be true about $p'(4)$?