[Math] Proving minima, maxima, and saddle points don’t exist.

calculusmultivariable-calculus

I've got the next function: $f(x,y)=x^3y^3$ where $x,y\in \mathbb R$

I need to determine whether there is minima, maxima or saddle point.

Easily enough, after doing the partial derivatives

  1. $f_x'(x,y)=3x^2y^3$
  2. $f_y'(x,y)=3y^2x^3$

I get the point $(0,0)$.

Now, how can i "officially" prove that that point isn't a minima, maxima or saddle point? Is it enough to show that hessian matrix gives 0? Do i need to show that the function can get higher [/lower] values using other points on the function?

Best Answer

Use the definition of maxima/minima. Try to find an open neighborhood $U$ containing $(0,0)$ s.t. for all $(x,y)\in U$ you have either

$$f(x,y)-f(0,0)>0 $$

or

$$f(x,y)-f(0,0)<0 $$

I supposed that the domain of $f$ is the whole $\mathbb R^2$. The answer to your question is then equivalent to study either

$$x^3y^3>0 $$

or

$$x^3y^3<0 $$

With an $(x,y)$ plot and focusing on what happens around $(0,0)$ you can arrive quite easily to the answer (=negative one, i.e. there exists no such neighborhood $U$, so $(0,0)$ cannot be either a max or a min).

Edit: I add the plot of $f$

Wolfram plot and contour plot of $f$ around $(0,0)$

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