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Put $\xi =x$ and $\eta =y-1$. The polynomial becomesThere are no terms of degree one or two, so the gradient and the entire Hessian matrix vanish at $(0,1)$. It is a critical point with singular Hessian matrix. The lowest nontrivial approximation isunderstood as the fourth-degree leading term rather than a uniform ratio asymptotic along its zero directions.

On the path $\eta =0$ the exact value is $-{\xi }^4<0$ for $\xi \ne 0$. On the path $\eta =-2\xi$ it is ${\xi }^4(1-2\xi {)}^2>0$ for sufficiently small nonzero $\xi$. Both paths approach the point, where $g=0$. It is neither a local maximum nor a local minimum; it is a degenerate saddle point of a scalar function. The higher-order saddle test supplies the conclusion the singular Hessian matrix cannot supply.