This is part of the proof of the differential criterion for convexity. $\Delta$ is actually the difference between the cord and the function $f$ at the point $x_0=px+qy$. If we expand $(6.8)$, we can see it is true. I do not see how do we get $(6.9)$, could any one give me some insight? I tried to look at the area, but didn't get it.
We know that $\Delta \ge 0$,
\begin{align}p\int_x^{px+qy}f'(u) \, du &\le q \int_{px+qy}^y f'(u) \, du \\& \le q\int_{px+qy}^y f'(px+qy) \, du \\ &= qf'(px+qy)[y-(px+qy)]\\ &= qf'(px+qy)[py-px] \\ &= pq(y-x)f'(px+qy)\end{align}