The identification of breast cancer patients for whom chemotherapy could prolong survival time is treated here as a data mining problem. This identification is achieved by clustering 253 breast cancer patients into three prognostic groups: Good, Poor and Intermediate. Each of the three groups has a significantly distinct Kaplan-Meier survival curve. Of particular significance is the Intermediate group, because patients with chemotherapy in this group do better than those without chemotherapy in the same group. This is the reverse case to that of the overall population of 253 patients for which patients undergoing chemotherapy have worse survival than those who do not. We also prescribe a procedure that utilizes three nonlinear smooth support vector machines (SSVMs) for classifying breast cancer patients into the three above prognostic groups. These results suggest that the patients in the Good group should not receive chemotherapy while those in the Intermediate group should receive chemotherapy based on our survival curve analysis. To our knowledge this is the first instance of a classifiable group of breast cancer patients for which chemotherapy can possibly enhance survival.