We reinvestigate decisions of a Cournot duopolist on process R&D and production by relaxing the restrictions on parameters that satisfy the second-order conditions of profit maximization. Returning to the firms' reaction functions derived from the profit maximization, this paper not only analyzes the case of corner solution but also suggests that restricting the values on parameters, as usually done in the literature, is unnecessary. In addition, we show that even in the symmetric firm case, the symmetric equilibrium is not necessarily stable, whereas a stable solution is not necessarily symmetric. Moreover, given sufficiently high production cost, process R&D could turn one of the two symmetric firms into a monopolist in the output market.