Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 19 / a / Solution

At a regular constrained stationary point, the change in $f$ along every allowed tangent direction must vanish. Thus its gradient is normal to the constraint curve, giving the Lagrange multiplier equationsprovided $\nabla g\ne 0$. Solve these equations for $x,y,\lambda$, then classify the candidates using the constraint or a constrained second-derivative test. Points where the constraint gradient vanishes require separate analysis.

For $n$ variables and $r$ regular independent constraints $g_j=c_j$, use one multiplier per constraint:There are $n+r$ scalar equations in the $n+r$ unknowns. Independence of the constraint gradients is the regularity condition; boundary points must be treated separately when present.