This study used task-based interviews to examine students' reasoning about multivariable optimization problems in a volume maximization context. There are four major findings from this study. First, formulating the objective function (i.e. the function whose maximum or minimum value(s) is to be found) in each task came easily for 15 students who participated in the study. Second, most of the students successfully used the Lagrange Multiplier method or the Substitution method (i.e. eliminating one of the variables from the objective function, finding partial derivatives, and solving the simultaneous equations that result) to find the critical points of the objective function in each task. Third, verifying/justifying extrema (i.e. showing that the dimensions students found in each task are dimensions of largest volume object) was problematic for a majority of the students. Fourth, solving systems of nonlinear equations to find critical points of the objective function in each task was particularly challenging for nearly half of the students. This study has characterized students' difficulties when solving multivariable optimization problems. Further research is needed to explore instructional approaches instructors could adopt to help students develop a robust understanding of multivariable optimization problems and the methods used to solve these problems.