# Critical points of multi-variable function.

I have the expression:

$$(x^3-y^2 )(x-1)$$

and have to find the critical points and their nature. The points found are:

$$(1,1) =$$ saddle point

$$(1,-1) =$$ saddle point

$$(\frac 3 4 ,0) =$$ global minimum

$$(0,0) =$$ inconclusive, however, when I take $$0.1$$ and $$-0.1$$ as $$2$$ points slightly higher or lower to find the nature of the point, I get a value positive or negative value for both the $$fxx$$ and the Hessian Determinant i.e. $$D < 0$$ at $$0.1$$ and $$D > 0$$ at $$-0.1$$.

I understand, for a single variable function, that two different signs for values higher or lower than the points, would mean the point $$(0,0)$$ is an inflection point. But since it is a multi-variable function, how can I determine the nature of $$(0,0)$$ since the Hessian Determinant must be $$< 0$$ for the points to be saddle points?

Note that the section of the surface over the line $$y = ax$$ is $$z = (x^3 - a^2x^2)(x-1) = x^2(x - a^2)(x - 1)$$ Near $$(0,0)$$ this is approximately $$z = a^2x^2$$. If $$a$$ is any non-zero value, this is a parabola, opening upward, with $$x = 0$$ as the apex. I.e., if you leave $$(x,y) = (0,0)$$ in any direction other than along the lines $$x = 0$$ or $$y = 0, z$$ is going to increase. Setting $$x = 0$$ also gives you the parabola $$z = y^2$$, so the same is true for travel along the $$y$$-axis.

But when $$y = 0$$, the section becomes $$z = x^3(x - 1)$$, which near $$x = 0$$ looks like $$z=-x^3$$, increasing in the negative $$x$$-direction, but decreasing in the positive $$x$$-direction.

Hence $$(0,0)$$ is a saddle point, since it decreases in one direction and increases in others, though it does not have the classic saddle shape.