Monopolist production with a perfectly inelastic demand curve?

Question

The monopoly equates MR and MC then matches that point to the demand curve. But, what if the demand curve doesn't exist at that point? They don't produce anything at all?

Answer 1

They will produce at the highest price they can, since quantity demanded will not change in response to a change in output.

Answer 2

For perfectly inelastic demand curve (vertical) unregulated profit-maximizing monopolist will just charge maximum price (maximizing TR), thus quantity will be constant in this case, variable is only price.