I seem to recall my teacher telling us about the necessary and sufficient conditions while finding the maxima/minima of functions. However, I can no longer find those conditions in my booklet and even on the internet. Can someone please tell me about them?
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This applies to local extrema only.
Given differentiable $F(x,y),$ a necessary condition for a local extreme to occur at a point $(x_0,y_0)$ is that the gradient satisfies $\nabla F(x_0,y_0)=0.$ (that is, both partial derivatives must vanish at $(x_0,y_0).$
A sufficient condition for a local extreme is that $\nabla F(x_0,y_0)=0$ and the Hessian $$ \left[ \begin{array}{cc} f_{xx}(x_0,y_0) & f_{xy}(x_0,y_0) \\ f_{xy}(x_0,y_0) & f_{yy}(x_0,y_0) \end{array}\right]$$ satisfies one one of:
1) $f_{xx}(x_0,y_0) f_{yy}(x_0,y_0) - [f_{xy}(x_0,y_0)]^2 \neq 0$ and $f_{xx}(x_0,y_0) + f_{yy}(x_0,y_0)>0$ (Local Min)
2) $f_{xx}(x_0,y_0) f_{yy}(x_0,y_0) - [f_{xy}(x_0,y_0)]^2$ and $f_{xx}(x_0,y_0) + f_{yy}(x_0,y_0)<0$ (Local Max)
matt biesecker
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I think it should be emphasized that, as you wrote, these are local extrema. I'm not aware of any general methods of finding the global ones. – Matias Heikkilä May 13 '15 at 17:14
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@MatiasHeikkilä. Thanks. For constrained problems with reasonably strict assumptions on the constraints, there are the KKT conditions. – matt biesecker May 13 '15 at 17:21
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1That sufficient condition, is also a necessary condition? – FCardelle Dec 27 '21 at 10:33
