We present a method for studying phase transitions in the largeN limit of matrix models using matched solutions of Whitham hierarchies. The endpoints ofthe eigenvalue spectrum as functions of the temperature are characterized both as solutionsof hodograph equations and as solutions of a system of ordinary differential equations. Inparticular we show that the free energy of the matrix model is the quasiclassicalτ-function of the associated hierarchy, and that critical processes in which thenumber of cuts changes in one unit are third-order phase transitions described byC1 matched solutions of Whitham hierarchies. The method is illustrated with theBleher–Eynard model for the merging of two cuts. We show that this model involves alsothe birth of a cut.