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question about critical points...

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Hi, I have been unable to find this in my textbook. So say I have \(f(x,y)\) at the point \((a,b)\) and \[\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\]
If I say \[\Delta(x,y)=\frac{\partial^2 f}{\partial x^2}\cdot\frac{\partial^2 f}{\partial y^2}-\left(\frac{\partial^2f}{\partial x\partial y}\right)^2\]
then if \[\Delta (a,b) > 0\quad \text{and}\quad \frac{\partial^2f}{\partial x\partial y}>0\] I have a relative maximum. And if\[\Delta (a,b) > 0\quad \text{and}\quad \frac{\partial^2f}{\partial x\partial y}<0\]I have a relative minimum.

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Other answers:

If \(\Delta (a,b) < 0\) I have a saddle point. And If \(\Delta (a,b) = 0\) I can't draw any conclusions. So I have two questions. The first one (most important) is what if \[\frac{\partial^2f}{\partial x\partial y}=0\] Then how do I know if this is a maximum or a minimum? The second question (less important for now), is why does this work!?!?
sorry the mixed partial derivatives are also evaluated at (a,b)

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