Question

Math question

Answer 1

@Michele_Laino Please help!

Answer 2

I think that it is an optimization problem, and in that case we have to apply the Kuhn-Tucker theorem

Answer 3

Yeah it's just solving systems of linear equations and inequalities

Answer 4

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

Answer 5

I'm sorry for that!

Answer 6

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)

Answer 7

is this linear programming problem

Answer 8

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

Answer 9

Right, so the maximum is at (-3, 2)??

Answer 10

well, look at the points of intersection again:

https://www.desmos.com/calculator/hq5cb8uttb

Answer 11

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 \(126\)

Answer 12

@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

Answer 13

Oh ok thanks. That makes sense.