For and example with positive marginal cost, see here:   • Bertrand Competition: Differentiated ...  

This video solves a problem based on Bertrand competition with differentiated goods. I have another video that reviews Bertrand competition with identical goods. IMPORTANT NOTE about Marginal Revenue in this video:

In Bertrand, a price competition model, the expression for MR is dTR/dP rather than the usual dTR/dQ. Likewise, if you setup up a profit function for each firm in this example, you would take the derivative of profit with respect to price, so the marginal profit expression is dProfit/P, not the usual dProfit/dQ. If you are still not convinced, see Microeconomics (page 467, including footnote #9) by Goolsbee, Levitt, and Syverson.