Why will a monopolist never choose to operate on the inelastic portion of its average revenue curve?

The average revenue curve of a monopolist is considered as the demand curve, which is negatively sloped. The average revenue curve is the ratio of total revenue to the quantity of output. The average revenue curve shows the price elasticity of demand. The level of output at which marginal revenue is zero, the demand for the monopolist?s good is unitary elastic; whereas, output at which marginal revenue is positive but decreasing, the demand for good is elastic.

The lower half region of the average revenue curve shows the inelastic demand for the monopolist?s good, where the marginal revenue becomes negative. The monopolist always charges a higher price even if lesser quantity of a good is sold because the higher price of good can easily compensate the loss incurred in the total revenue as fewer quantities will be sold. When the marginal revenue becomes negative, then the total revenue will start falling. So, a monopolist can't operate in this inelastic region as the total revenue will decline.