At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
@Michele_Laino Please help!
I think that it is an optimization problem, and in that case we have to apply the Kuhn-Tucker theorem
Yeah it's just solving systems of linear equations and inequalities
yes! Nevertheless I'm not good with that theorem, since I don't have studied it, I only saw it from my mathematician friends, when I was at university
I'm sorry for that!
All you need here is the fundamental theorem of linear programming, which tells us that for a linear objective function constrained by linear inequalities we reach extreme values at the extreme points (I.e. vertices) of the feasible region (the polygon you get from the region that satisfies all the inequalities)
is this linear programming problem
So plot those inequalities and look at where they intersect; you should get a feasible region in the shape of a triangle. Test the three corner points of your triangle in the objective function and whichever is largest yields the maximum objective value subject to those constraints
Right, so the maximum is at (-3, 2)??
well, look at the points of intersection again: https://www.desmos.com/calculator/hq5cb8uttb
we have \((3,6),(4,2),(9,9)\) as our vertices for the feasible region, and now we test our objective here: $$f(3,6)=9(3)+5(6)=57\\f(4,2)=9(4)+5(2)=46\\f(9,9)=9(9)+5(9)=126$$ so the maximum is at \((9,9)\) for which \(f\) attains the value \(126\)
@Michele_Laino Kuhn-Tucker conditions are for optimality of solutions to nonlinear optimization problems and is overkill for linear problems; the above method or the more efficient simplex algorithm for more complicated polytopes is ideal
Oh ok thanks. That makes sense.