= 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 gives
$$
f_xg_y-f_yg_x=\lambda g_xg_y-\lambda g_yg_x=0.
$$
Thus the required <Lagrange multiplier determinant condition in two variables> is
$$
\boxed{\det\begin{pmatrix}f_x&g_x\\f_y&g_y\end{pmatrix}_{(a,b)}=0.}
$$
If 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).
Back to article page
