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Determine and classify all the critical points of $f(x,y)= x^2 y^3$.

I first differentiate w.r.t $x$ and $y$ and set the equations equal to zero. But that would means all the points $(0,y)$ and $(x,0)$ are critical points. How could that be possible and how to classify whether the point is a local max/min or saddle point in this case?

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  • $\begingroup$ Sometimes it is a good idea to actually examine the graph. I did that and here is the result. math.uri.edu/~bkaskosz/flashmo/graph3d It may help to understand the answers below. It does show for example that at the Origin, there is a horizontal tangent plane. $\endgroup$ – imranfat Dec 2 '13 at 18:58
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The idea of a critical point and a critical value needs not have any idea of a graph, e.g. $z=f(x,y)$.

As you rightly say, the critical points are given by $2xy^3=3x^2y^2=0$, and this happens when $x=0$ or $y=0$. The $x$-axis and the $y$-axis for the set of critical points. Notice also that $f(x,0)=f(0,y)=0$, meaning that $0$ is a critical value.

What this tells us is that $f^{-1}(0) = \{(x,y):f(x,y)=0\}$ fails to be a smooth curve in a neighbourhood of each of its points. Indeed, the $x$-axis and $y$-axis, thought of as a single set, have a node at the origin. Moreover, $f^{-1}(c) = \{(x,y) : f(x,y)=c\}$ will be a smooth curve in a neighbourhood of each of its points when $c \neq 0$ since $0$ is the only critical value, i.e. $c \neq 0$ are all regular values.

It's very difficult to classify these critical points because they are not isolated. The standard way to classify critical points is to look at their Milnor Number: $$\mu = \dim\left(\frac{\mathbb{R}[x]}{\langle 2xy^3,3x^2y^2 \rangle}\right)$$ However, in this case, $\mu = \infty$. These are very, very badly behaved critical points. For example, non-degenerate, Morse-type critical points: $f(x,y)=x^2 \pm y^2$ have $\mu = 1$. Then the $A_k$-series $f(x,y) = x^2 \pm y^{k+1}$ have $\mu = k$. There are other series, e.g. $D_{\mu}, E_6,E_7$ and $E_8$. These all have finite Milnor Number.

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You are correct. These are not isolated critical points, but you can still ask whether $f(a,0)\ge f(x,y)$ (or $\le$ or neither) for all $(x,y)$ near $(a,0)$, and whether $f(0,b)\ge f(x,y)$ (or $\le$ or neither) for all $(x,y)$ near $(0,b)$.

HINT: Your answer will depend on whether $b$ is positive, zero, or negative, but not on the sign on $a$. Why?

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