Why does MC cut both AVC and ATC curves at their minimum?

Can anyone explain?

Well, in words: Read Carefully

For as long as AC is decreasing with each additional unit of output, the cost of producing an extra ("marginal") unit of output must be less than the AVERAGE cost at the previous level of output, so "total cost/output" gets smaller so AC gets smaller.
Therefore for as long as AC is decreasing, MC is below AC.

The reverse is also true; for as long as AC is increasing with each additional unit of output , the cost of producing each marginal unit of output must be greater than the AVERAGE cost at the previous level of output, so "total cost/output" gets bigger, so AC gets bigger.
Therefore for as long AC is increasing, MC is above AC.

So when AC has negative gradient, MC<AC;
when AC has positive gradient MC>AC.
So when AC gradient=0 i.e. the "turning point" in maths-speak, MC cuts AC.

It's just logic. :biggrin: