When demand is inelastic, a monopolist has an incentive to raise price because it will raise revenue. Quantity produced and demanded will fall but the demand is inelastic so % increase in price exceed the % decrease in quantity so that overall the % change in revenue is positive. Till the demand is inelastic, a profit maximizing monopolist has an incentive to raise price and and reduce quantity. It is when demand becomes either unitary elastic or relatively elastic that the monopolist stabilize.
Next, see that since MC is rising for a larger range of output, reduction in quantity will reduce cost, and we have seen that reduction in quantity (when it raises the price) will raise revenue. So revenue rises and costs fall as quantity is reduced. This will continue till the difference is maximized which happens only in the elastic portion of demand.
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