Lagrange multipliers

The method of Lagrange multipliers finds constrained extrema from the geometric fact that at a constrained extremum, the objective's gradient and the constraint's gradient point along the same line: \nabla f=\lambda\nabla g. In practice this expands to two scalar equations, f_x=\lambda g_x and f_y=\lambda g_y, joined by the constraint g=k itself — three equations in x,y,\lambda. Solving the system produces candidates for the extremum, not guaranteed extrema; comparing f's value across every solution (invoking the multivariable extreme value theorem when the constraint set is closed and bounded) is still the last step.