Figures 7.c and d display the behaviour of the buffer index for two values of r C . It is clear that exiting the 5 roots domain leads to buffer index curves with a single maximum while at the same time the values of ∆pK c a decreases as r ∆pKa increases in the 3 roots domain, leading to narrower flat region in the buffer index curve. Hence, the optimal solution must be found on the 5-3 boundary. Figure .e and f display respectively the buffer index and its derivative along the critical line for ∆pK a = ∆pK c a . At low values of r C , beta actually displays several shoulder that tend to disappear for r C > 1.5. In order to optimize more precisely, it becomes necessary to proceed as above, by examining the root space in detail.