The function belongs to where is the open set .
On the other hand, also belongs to and =1" alt="\textrm{rank}(\;\nabla g(x_1,\ldots,x_n)\=1" /> for all (maximun rank).
According to the multipliers Lagrange theorem if has a local maximum or minimum at there exists such that is critical point of
Verify that such a point does not exist. This means that the absolute maximum an minimum value for is in the boundary of the compact set:
because is continuous.
There are only two variables and , and the function has no calculus like minimum or maximum in the interior of the feasible region.
As the feasible region is closed there is a minimum/maximum on the boundary and it can be found by examination of the function on the boundary.
CB
First of all, you should show some work. Look rule #11 here:
http://www.mathhelpforum.com/math-he...ng-151424.html
I propose you the following: consider (then, you'll analyze what difficulties you can find in the general case).
(i) Prove that there are no critical points in for:
(ii) Find the absolute maximum and minimum of
" alt="f(x,y,z)=\dfrac{x}{x+y},\quad \partial (K) \equiv \begin{Bmatrix} a_1\leq x \leq b_1\\ a_2 \leq y \leq b_1\\a_3\leq 1-x-y \leq b_3 \end{matrix}\quad (\; [a_i,b_i]\subset (0,1)\ " />
Draw in the plane the set . According to the relations of with , analyze the different kind of boundaries you can find.
P.S. Have you seen the problem in a book? . In what a context?. Is it homework?.
Dear Prof. Fernando Revilla,
First of all thank you very much for every thing.
My problem is neither in a book nor a homework. Nowadays I am working on a paper and I countered this optimization problem. I think if I was able to sove it, then I would get some results concerning my research. Therefore, I first introduced the problem in a general case without determining the function and the constraint.
Best Regards.