A business could sell its products at a loss for a short duration of time if doing so could improve the chances of the business making higher profits in the future.
This can be the case if the business is a new entrant and there are already many competitors to deal with. The low prices could be used to make people aware about its products, use them and make them switch from the players already in the market.
This is not something that can be sustained over a long time and the business would soon have to increase the price of the products to a sustainable cost and hope the customers that have been acquired continue to stick with it.
Another reason why a business would sell its products at below cost would be to force smaller competitors to shutting their operation. This can only be done by very large businesses with sufficiently large reserves who want to expand their consumer base and enter new markets. The strategy would work if the pre-existing players in the new markets cannot compete with the low rates and do not have reserves to sustain loss for a small time duration.
Yes. There are circumstances in which a business should (at least in the short term) sell its product for less than the total cost of making the product.
A business should do this when the price that it can charge for its product is higher than the average variable cost of the product but lower than the average total cost. In such a situation, the company will lose less money by selling at a loss than it would by shutting down.
However, if the price that the company can charge drops below the average variable cost for the product, the company should shut down because producing the item loses them more money than shutting down would.