I suggest you plot a logarithmic diagram of your system, i.e. pH = f(log Ci), where Ci is the concentration of each protolyte in the solution. The buffer capacity can then be plotted as
pH = f(log(β/ln10)).
If you have plotted your logarithmic diagram properly, you can directly plot the line for log(β/ln10).
I will give you an example:
Suppose we have a weak acid HA with the pka = 4.8.
The line for log(β/ln10) goes from (0; 0) along the line of log [H3O+] until it reaches the crossing with the line for log [A-]. Here it bends upwards 0.3 log C units above the crossing point. It then follows the line for log [A-] close to pH = pka. Here it bends down, passing pH = pka, 0.6 log C units below the line for log Ctot. Then the line for log(β/ln10) will follow the line for log [HA] until it reaches the crossing point with the line for
log [OH-]. Here the line for log(β/ln10) bends up again and passes the crossing point 0.3 units higher. Then the line for log(β/ln10) will follow the line for log [OH-] to pH = 14.
From this example, we can immediately identify a local maximum for the buffer capacity, i.e. at pH = pka. From the diagram, we can also identify two local minima, which will be at about pH 3 and close to pH 9. We can also directly estimate the maximum buffer capacity from the diagram.
Logarithmic diagrams are excellent tools for this kind of problems. If you understand the rules, you can plot the buffer capacity (log(β/ln10) directly in the diagram without any calculations.
I tried to copy in a logarithmic diagram of the example above, but it did not work.