When you say "competitive firm" I'm not sure if you mean a firm in perfect competition or one in monopolistic competition.
If you mean perfect competition, the only way this statement is wrong is in the fact that it says that the firm sets its price. This implies that the firm has control over what price it charges. In reality the firm is a price taker and does not control its price.
If you mean monopolistic competition, then this statement is always false. In this market structure, a firm produces the *quantity* where marginal revenue = marginal cost. But the price is higher than marginal cost because the demand curve is to the right of the MR curve.
In case of perfect competition (adjustment amount), the price is fixed so as to be equal the marginal costs to achieve maximum profit.
In case of normal monopoly, there is an area where increasing marginal cost intersects the decreasing curve of the sales.At this point of intersection, is the combination of the quantity offered and the price obtained that maximizes the total gain of the monopolist. This price is, ceteris paribus, higher than the price set by those who adjust the quantity and the amount offered is less than in case of perfect competition.
In case of "natural monopoly" average costs are decreasing by quantity.In this case there isn't a point of intersection between marginal costs and average costs, because marginal costs are always below average costs. Therefore, such a monopoly can not cover its costs with marginal costs, so it must be set a price at least equal to average costs.
Only when the marginal costs exceed average costs, the price set can be equal to the marginal costs,so that to be covered all costs.