Finding the extremal point of a function I have to find and classify all extremal points of a few problems and I am finding it difficult as I don't know how to even start. I can't find any examples of what to do. So if someone could help with one question then I can use that to figure out the rest. One of the questions is  
$$f(x, y) = 5xy − 7x^2 + 3x − 6y + 2.$$
Thanks in advance.
 A: In general, when having real functions $ f: \mathbb{R}^m \rightarrow \mathbb{R}^n $, which are continuous and twice differentiable, to find their extreme values, you may follow the following strategy:


*

*Find the gradient of $f$, and study its critical points (i.e. points where  $\nabla f =0$). Note, it may happen that $f$ has no critical points, and hence no extreme points in the open domain it is defined on. 

*Find the Hessian matrix of  $f$ at each of the critical points. If the Hessian matrix is positive definite, then the corresponding critical point is a local minimum, and if it is negative definite then the corresponding  critical point is a local maximum. If the Hessian matrix associated to a critical point is neither positive nor negative definite then your point is neither a local min nor a local max. 


Some Special cases:


*

*If your function is convex (resp. concave) then your function has a unique global min (resp. max).

*If your function is defined on a closed domain, the you may study  optimization first in the interior of your closed domain (thus considered as optimization on an open domain), then study optimization of $f$ on the boundary, using for example Lagrange multipliers. 
Hope this helps you, try for the above defined function in your question, and I am ready for any clarification. 
