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richyw Group TitleBest ResponseYou've already chosen the best response.0
Hi, I have been unable to find this in my textbook. So say I have \(f(x,y)\) at the point \((a,b)\) and \[\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\]
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
If I say \[\Delta(x,y)=\frac{\partial^2 f}{\partial x^2}\cdot\frac{\partial^2 f}{\partial y^2}\left(\frac{\partial^2f}{\partial x\partial y}\right)^2\]
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
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.
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
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!?!?
 one year ago

richyw Group TitleBest ResponseYou've already chosen the best response.0
sorry the mixed partial derivatives are also evaluated at (a,b)
 one year ago
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