f(x)-(8/9)x(12-x)=(8/9)(12x-x^2). If we differentiate we get: f'(x)=(8/9)(12-2x), and when f'(x)=0, we have a maximum or minimum value of f(x). So, when 12-2x=0, 6-x=0 and x=6, and f(x)=(8/9)6(12-6)=32. If we choose values of x near to x=6, say, 5 and 7 we get: f(5)=f(7)=31.11, so f(6)=32 is a maximum, the maximum number of customers.
Another way to solve this is to plot the graph which is a parabola with vertex at (6,32).