Angle dependent magnetothermal conductivity experiments on CeCoIn5 indicate that this compound is a d(x)(2)-y(2)-wave superconductor. In this study, the low-temperature behavior of the upper critical field is measured in a single crystal of CeCoIn5 along the directions (H) over right arrow parallel to(a) over right arrow and (H) over right arrow parallel to(c) over right arrow. The data are compared with model calculations of the upper critical field in a d(x)(2)-y(2)-wave superconductor. It is found that the observed H-c2(T) along (H) over right arrow parallel to(a) over right arrow is consistent with a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state at low temperatures, T<0.7 K, whereas for (H) over right arrow parallel to(c) over right arrow the FFLO state appears to be absent in CeCoIn5. Furthermore, it is predicted that the quasiparticle density of states in the FFLO state exhibits a complex peak structure which should be observable by scanning tunneling microscopy.