Vector Calculus | Part IA, 2005

Explain, with justification, the significance of the eigenvalues of the Hessian in classifying the critical points of a function f:RnRf: \mathbb{R}^{n} \rightarrow \mathbb{R}. In what circumstances are the eigenvalues inconclusive in establishing the character of a critical point?

Consider the function on R2\mathbb{R}^{2},

f(x,y)=xyeα(x2+y2)f(x, y)=x y e^{-\alpha\left(x^{2}+y^{2}\right)}

Find and classify all of its critical points, for all real α\alpha. How do the locations of the critical points change as α0\alpha \rightarrow 0 ?

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