Why does the SMC curve cut the AVC curve at the minimum point of the AVC curve?

Looking at the graph given below we can say that when AVC falls SMC falls at a greater rate than AVC. When AVC starts rising SMC is rising at a rate faster than the AVC. Hence only when AVC is constant and at its minimum point, here at P, that is only when SMC is qual to AVC. Therefore, AVC cuts SMC at its minimum point.

Source: NCERT