Differential Equations | Part IA, 2008

Consider the function

f(x,y)=x2+y212x4bx2y212y4f(x, y)=x^{2}+y^{2}-\frac{1}{2} x^{4}-b x^{2} y^{2}-\frac{1}{2} y^{4}

where bb is a positive constant.

Find the critical points of f(x,y)f(x, y), assuming b1b \neq 1. Determine the type of each critical point and sketch contours of constant f(x,y)f(x, y) in the two cases (i) b<1b<1 and (ii) b>1b>1.

For b=1b=1 describe the subset of the (x,y)(x, y) plane on which f(x,y)f(x, y) attains its maximum value.

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