Paper 2, Section II, A

(a) State how the nature of a critical (or stationary) point of a function $f(\mathbf{x})$ with $\mathbf{x} \in \mathbb{R}^{n}$ can be determined by consideration of the eigenvalues of the Hessian matrix $H$ of $f(\mathbf{x})$, assuming $H$ is non-singular.

(b) Let $f(x, y)=x y(1-x-y)$. Find all the critical points of the function $f(x, y)$ and determine their nature. Determine the zero contour of $f(x, y)$ and sketch a contour plot showing the behaviour of the contours in the neighbourhood of the critical points.

(c) Now let $g(x, y)=x^{3} y^{2}(1-x-y)$. Show that $(0,1)$ is a critical point of $g(x, y)$ for which the Hessian matrix of $g$ is singular. Find an approximation for $g(x, y)$ to lowest non-trivial order in the neighbourhood of the point $(0,1)$. Does $g$ have a maximum or a minimum at $(0,1)$ ? Justify your answer.

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