Why does the SMC

The SMC curve always intersects the AVC curve at its minimum, because at the minimum point of AVC the SMC is below AVC.

Both SMC and AVC fall but SMC Falls at a faster rate than AVC. Say the minimum point is K where AVC is equal to SMC beyond this point the AVC and SMC both rises but SMC Rises at a faster rate and is above AVC therefore the only point where SMC and AVC are equal is where is SMC intersects AC that is the minimum point K

Let's understand it with help of graph –

Cost Schedule

GRAPH