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What do we do when the second derivative test fails?

For example, I'm asked to find all the critical points of the function $f(x,y)=x^{2013}−y^{2013}$ and determine the nature of the critical points. The critical point that I have found is at (0,0), but I'm unable to determine its nature as the second derivative test fails here.

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1  
do $x^3 - y^3$ first. –  Will Jagy Mar 21 at 20:20

4 Answers 4

Let $y=0$ and see what your function looks like. Then let $x=0$ and see what the function looks like.

Alternatively: Here's a very similar function (in terms of behavior around $(0,0)$): $g(x,y) = x^3-y^3$. It might help to see what this does if it's not clear to you what $f$ does. See here if you still have difficulty.

Here's a more rigorous argument for why it is a saddle point: Let $y=cx$. Then

$$f(x,y) = x^{2013}-c^{2013}x^{2013} = (1-c^{2013})x^{2013}.$$

For any value of $c$, this does not have concave-up nor concave-down behavior. Hence it cannot be a local min or max and hence is a saddle point.

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Can be seen easily without second derivatives that (0,0) is a saddle point because $f$ takes values $>f(0,0)$ and $<f(0,0)$ arbitrarily near of (0,0):

For $(x,y)=(\epsilon,0)$, $\epsilon>0$: $f(x,y)=\epsilon^{2013}>0$.

For $(x,y)=(0,\epsilon)$, $\epsilon>0$: $f(x,y)=-\epsilon^{2013}<0$.

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Are you sure about this? The definition of saddle points I am aware of is that the function is concave up in one direction and concave down in the other. This function does not display such behavior. See the edit to my comment. –  Cameron Williams Mar 21 at 19:13
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"... a saddle point is a point in the domain of a function that is a stationary point but not a local extremum." (mathworld.wolfram.com/SaddlePoint.html, en.wikipedia.org/wiki/Saddle_point). –  Martín-Blas Pérez Pinilla Mar 21 at 19:15
    
Oh. Well then I guess I'll edit my comment to reflect this. I guess my multivariable calculus professor wasn't the best. –  Cameron Williams Mar 21 at 19:16

Hint:

  • Take into consideration higher-order derivatives.
  • Note the parity of the first non-zero derivative.
  • What are the similarities among $x^3, x^5, x^7,\ldots$ and similarities among $x^2,x^4,x^6,\ldots$ (e.g. how the graphs would look like, and what is the parity of the first non-zero derivative)?
  • You can read more about it here.

I hope this helps $\ddot\smile$

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Note that $f(x,y) > 0$ whenever $x>0$ and $y<0$. Moreover, whenever $x<0$ and $y>0$ we have $f(x,y)<0$. Thus, as $f(0,0)=0$, we have points $(x_1,y_1)$ and $(x_2,y_2)$ in every neighborhood of $(0,0)$ such that $f(x_1,y_1) < f(0,0) < f(x_2,y_2)$. As $f$ is defined on $\mathbb{R}^2$ and hence on the just constructed points, we see that $(0,0)$ satisfies the definition of a saddle point of $f$.

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