Past exam of the options I course of the University of Oxford / 2006 / F / F1

The rows of the Jacobian matrix $Dg(a)$ are $\nabla g_1(a{)}^T,\dots ,\nabla g_k(a{)}^T$. Rank $k$ means precisely that these $k$ gradients are linearly independent. The Lagrange multiplier condition is that real numbers ${\lambda }_1,\dots ,{\lambda }_k$ exist withGeometrically, $\nabla f(a)$ is perpendicular to the tangent space of the constraint level set, whose tangent vectors form $\ker Dg(a)$. The full-rank hypothesis ensures this level set is a smooth local constraint manifold.

The constraint set $S^1\times S^2$ is compact, so the continuous objective attains a maximum and minimum. The two constraint gradients are nonzero and supported on separate coordinates, so they are linearly independent everywhere on it. The Lagrange multiplier equations areBoth positive and negative values occur. At a maximizing or minimizing point the objective is therefore nonzero, so $x_1\ne 0$, $y_2-y_3\ne 0$, and both $\lambda ,\mu$ are nonzero. The equations imply $x_2=0$, $y_1=0$, and $y_3=-y_2$. The constraints then give $x_1=\pm 1$, $y_2=\pm 1/\sqrt{2}$.

Consequently, writing $s=\pm 1$, the complete list of extrema isFor an independent global check, Cauchy-Schwarz inequality gives $|x_1(y_2-y_3)|\le |x_1|\sqrt{2(y_2^2+y_3^2)}\le \sqrt{2}$, with equality exactly at the displayed points.