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vera_ewing

  • one year ago

Math question

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  1. vera_ewing
    • one year ago
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    @Michele_Laino Please help!

  2. Michele_Laino
    • one year ago
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    I think that it is an optimization problem, and in that case we have to apply the Kuhn-Tucker theorem

  3. vera_ewing
    • one year ago
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    Yeah it's just solving systems of linear equations and inequalities

  4. Michele_Laino
    • one year ago
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    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

  5. Michele_Laino
    • one year ago
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    I'm sorry for that!

  6. anonymous
    • one year ago
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    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)

  7. mathmath333
    • one year ago
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    is this linear programming problem

  8. anonymous
    • one year ago
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    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

  9. vera_ewing
    • one year ago
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    Right, so the maximum is at (-3, 2)??

  10. anonymous
    • one year ago
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    well, look at the points of intersection again: https://www.desmos.com/calculator/hq5cb8uttb

  11. anonymous
    • one year ago
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    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 \(126\)

  12. anonymous
    • one year ago
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    @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

  13. vera_ewing
    • one year ago
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    Oh ok thanks. That makes sense.

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