I currently have an multi-variable function of the form:
$F(x,y) = 1500x + 500y + 5x^2y $
The question I have to address is maximising the above function, taking into account the inequality constraint: $5000x + 2800y <= 400000$
How do you use Lagrange multipliers (or any other method per se) to solve this problem? I read the Wikipedia page on the topic (https://en.wikipedia.org/wiki/Lagrange_multiplier#Multiple_constraints), but was still unable to understand how to do the question. I also read other information on the issue (http://privatewww.essex.ac.uk/~wangt/Presession%20Math/Lecture%205.pdf) but am virtually clueless. Please help!!
There are several ways of handling those situations, there exists, for example, an extension of Lagrange Multipliers called the Karush–Kuhn–Tucker conditions (https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions). I am however not keen on using it unless you can't find a clever way of coming up with a solution. As you might have noticed in this context, the score function only has increasing components(i.e $1500x$ is increasing when $x$ is increasing same with 500$y$). So in order to maximize the function, you would like your coefficient $x$ and $y$ to be maximal which happens at the boundary. So your maximum occurs when you are at the equality!
One other way of solving this is by disjointing the set of constraint into two distinct constraints. Let $M=\{5000x + 2800y = 4000\}$ and $N=\{5000x + 2800y < 4000\}$. You look for a global extremum within $N$ and if no such point exist then the maximum of the function under the constraint lies in $M$.
Hope this helps!