I have a function $g(x,y,z)$, and $x+y+z=1, x\geq0,y\geq0,z\geq0$. Now I want to maximize $g$.
If I ignore the inequalities, then I can use lagrangian and can solve this thing for maximum. But I am not sure how to incorporate the inequalities in the system of equations.
Would greatly appreciate if you can name some methods which I can apply here.
You can replace $g(x,y,z)$ with $h(x,y)=g(x,y,1-x-y)$. Now you need to find the maximum of $h$ on $$S=[0,1]\times[0,1]$$ Which is a little easier. You know that if $h$ has a maximum in the interior of $S$, its partial derivatives will equal zero at that points. Then you simply check for possible maximums on the edges of $S$ (which is slightly annoying, since you must find the maximum of $h(0,y),h(1,y),h(x,0)$ and $h(x,1)$, but it is doable).