Why will the profit maximizing monopolist never operate on the inelastic portion of its demand curve?

A profit-maximizing monopolist will never operate on the inelastic portion of its demand curve because, in that range, increasing the price leads to a decrease in total revenue. Since demand is inelastic, a price increase results in a proportionally smaller decrease in quantity demanded, causing total revenue to fall. To maximize profit, the monopolist will only produce where demand is elastic, where price increases would lead to higher total revenue. Thus, operating on the inelastic portion would be counterproductive to profit maximization.