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 $$p'(4)$$?