No, a profit maximising firm will not produce in the range where the marginal cost is falling. This is because, at this range, his profit is not maximised. So he fixes his level of out output at the point where marginal cost equals marginal revenue. This can be explained with the help of a diagram.
As per the diagram, if the firm fixes output at the range where MC is falling his profit will not be maximised. Being profit maximising rm, he goes on producing OQ level of output corresponding to the point where MC=MR. This is at the rising part of MC. So his profit is maximised as shown in shaded area.
