Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 20 / Solution

At a regular constrained stationary point, the Lagrange multiplier equations are $f_x=\lambda g_x$, $f_y=\lambda g_y$. Eliminating the multiplier givesThus the required Lagrange multiplier determinant condition in two variables isIf the constraint gradient vanishes, this determinant is zero automatically, although the regular multiplier theorem cannot be applied. The condition is necessary and is not sufficient to classify the constrained point. This addresses the original unnumbered task before part (a).