The figure shows the graph of a function $f$ on $[0, 12]$. It is made of three straight pieces joining the points $(0, 0)$, $(3, 6)$, $(9, 0)$ and $(12, -3)$: a steep rise, a slow fall, and a dip below the axis. For $0 \le x \le 9$ let $\displaystyle F(x) = \int_x^{x+3} f(t)\,dt,$ the area under the graph over a window of width $3$ that slides along the axis. The window catches the most area when its two ends are at the same height, but the peak of $f$ need not sit in the middle of the window. What is the greatest value of $F(x)$ for $0 \le x \le 9$?

Fig. 1: The graph of a function f on [0, 12] made of three straight pieces: it rises from (0, 0) to (3, 6), falls to (9, 0) and goes on falling to (12, -3). The four corner points are marked with their coordinates. A light grid is shown. A window of width 3 on the x-axis is shaded under the graph in one sample position, from x = 5 to x = 8.