Regarding costs of production, can a firm ever be at a point that is not on the marginal cost curve? Explain

A profit-maximizing, price-taker firm always operates along its marginal cost curve. If output expands, we move rightward along the curve, and if it contracts, we move along the curve to the left. Since, by definition, a profit-maximizing price taker will produce the quantity where price (marginal revenue) equals marginal cost, the cost of producing the last unit will always be relevant, and you never move to a point that is not on the marginal cost curve.