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vera_ewing
 one year ago
Math question
vera_ewing
 one year ago
Math question

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vera_ewing
 one year ago
Best ResponseYou've already chosen the best response.0@Michele_Laino Please help!

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0I think that it is an optimization problem, and in that case we have to apply the KuhnTucker theorem

vera_ewing
 one year ago
Best ResponseYou've already chosen the best response.0Yeah it's just solving systems of linear equations and inequalities

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0yes! 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

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0I'm sorry for that!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0All 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)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0is this linear programming problem

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So 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

vera_ewing
 one year ago
Best ResponseYou've already chosen the best response.0Right, so the maximum is at (3, 2)??

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well, look at the points of intersection again: https://www.desmos.com/calculator/hq5cb8uttb

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0we 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 \(126\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@Michele_Laino KuhnTucker 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

vera_ewing
 one year ago
Best ResponseYou've already chosen the best response.0Oh ok thanks. That makes sense.
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