The large-N one-matrix model with a potential V(φ)=φ 2 /2+g 4 φ 4 /N+g 6 φ 6 /N 2 is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients r k (k=1,2,3,...) of the orthogonal polynomials at large N. We find that for g 6 /g 4 2 >1/2 there is no m=2 solution which can be expressed as a smooth function of k/N in the limit N→∞. This means that the assumption of smoothness of r k at N→∞ near the critical point, which was essential to derive the string susceptibility and the string equation, is broken even at the tree level of the genus expansion by adding the φ 6 term