3 Diodes-circuit analysis
All diodes are assumed to be ideal.

When Vi > 3V, D3 started conducting. 

When it started conducting, Vo = VA will be at \$(Vi + 3)/2\$ volts.

Which means, when Vi = 6V, VA = 4.5V. D1 does not stop conducting. It will stop conducting only at VA = 6V, i.e, when Vi = 9V. 

After 9V, only D3 will conduct. When it conducts, Vo = VB = \$(2/3)Vi\$ volts.

Which means, Vi should reach 30V, for D2 to start conducting. After that both D2, D3 will conduct upto 50V.

Your ranges [3,6] [6,20], [20,50] therefore are incorrect.

You can learn behavior of these circuits by a simulator. 

See here