Vector Calculus | Part IA, 2001

Explain, with justification, how the nature of a critical (stationary) point of a function f(x)f(\mathbf{x}) can be determined by consideration of the eigenvalues of the Hessian matrix HH of f(x)f(\mathbf{x}) if HH is non-singular. What happens if HH is singular?

Let f(x,y)=(yx2)(y2x2)+αx2f(x, y)=\left(y-x^{2}\right)\left(y-2 x^{2}\right)+\alpha x^{2}. Find the critical points of ff and determine their nature in the different cases that arise according to the values of the parameter αR\alpha \in \mathbb{R}.

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