Suppose that the average of a bunch of numbers is 10. If the next number you add to the list is 11, the average of the new bunch will be greater than 10; if the next number you add to the list is 9, the average of the new bunch will be less than 10.
Thus, if the marginal whatever curve is above the average whatever curve, the average whatever is increasing; if the marginal whatever curve is below the average whatever curve, the average whatever curve is decreasing. Furthermore. at a maximum value on the average whatever curve – where it is neither increasing nor decreasing – the marginal whatever curve crosses the average whatever curve from above to below; at a minimum value on the average whatever curve – where it is neither increasing nor decreasing – the marginal whatever curve crosses the average whatever curve from below to above.
All of this is true whatever whatever is. In this case, whatever is cost: when marginal cost is below average (total or variable) cost, total cost is decreasing, and when marginal cost is above average (total or variable) cost, total cost is increasing. Marginal cost equals average (total or variable) cost at the minimum point on the average (total or variable, respectively) cost curve.