Home > Analysis, Problem Solving, Undergraduate > Critical points and extremal points

Critical points and extremal points


Let f(x,y) be an infinitely differentiable function of two variables with a local minimum at the origin and no other critical points. Is it true that the minimum is global?

( A critical point is a point where both partial derivatives \frac{\partial f}{\partial x},\ \frac{\partial f}{\partial y} vanish. )
Answer: Counterexample:
\displaystyle f(x,y)=-\frac{1}{1+x^2}+\left( e^x+\frac{1}{1+x^2} \right) (3y^2-2y^3)

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