richyw 4 years ago question about critical points...

1. richyw

Hi, I have been unable to find this in my textbook. So say I have $$f(x,y)$$ at the point $$(a,b)$$ and $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0$

2. richyw

If I say $\Delta(x,y)=\frac{\partial^2 f}{\partial x^2}\cdot\frac{\partial^2 f}{\partial y^2}-\left(\frac{\partial^2f}{\partial x\partial y}\right)^2$

3. richyw

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.

4. richyw

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!?!?

5. richyw

sorry the mixed partial derivatives are also evaluated at (a,b)