Paper 2, Section II, A

Differential Equations | Part IA, 2010

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

(b) Let f(x,y)=xy(1xy)f(x, y)=x y(1-x-y). Find all the critical points of the function f(x,y)f(x, y) and determine their nature. Determine the zero contour of f(x,y)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)=x3y2(1xy)g(x, y)=x^{3} y^{2}(1-x-y). Show that (0,1)(0,1) is a critical point of g(x,y)g(x, y) for which the Hessian matrix of gg is singular. Find an approximation for g(x,y)g(x, y) to lowest non-trivial order in the neighbourhood of the point (0,1)(0,1). Does gg have a maximum or a minimum at (0,1)(0,1) ? Justify your answer.

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