Question

classify the critical points of f(x,y)=exp ax^2+by^2

Answer 1

first find the critical points by setting
\[\frac{\partial f}{\partial x}=0, \frac{\partial f}{\partial y}=0\]

Answer 2

\[\frac{\partial f}{\partial x}=2ax\exp(ax^2+by^2)=0\Rightarrow x = 0\\\frac{\partial f}{\partial y}=2ay\exp(ax^2+by^2)=0\Rightarrow y = 0\]
the only critical point is (0,0)

Answer 3

find all second-order partial derivatives at (0,0) of the function
f(x,y)=\[x ^{2} \tan^{-1} \frac{ y }{ x }-y ^{2}\tan^{-1} \frac{ x }{ y },x \neq0,y \neq0\]