Lagrange is best remembered for the Lagrangian function and Lagrange multiplier, which bear his name. Lagrange multipliers are used to locate multivariable maximum and minimum points subject to a constraint of the form g(x,y) = 0 or g(x,y,z) = 0.

He also made numerous contributions to the calculus of variations (which include optimization problems), calculus of probabilities, analytical mechanics, the theory of functions, and in differential and integral calculus.