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MathematicaSimultaneous Inequalities?

  1. Apr 22, 2010 #1
    Hey guys,

    I have some variables A,B, and C such that all must be greater then zero. A,B and C are all functions of K. I would like to find the range of values of K such that the 3 inequalities are satisfied.

    Is there a function built in to Mathematica that will do this? I could write a code, but I am kind of sick of coding right now :tongue2:

    If not..then code it is!
     
  2. jcsd
  3. Apr 22, 2010 #2

    Hepth

    User Avatar
    Gold Member

    Well, as a quick response I know Reduce can sort of find the limits if you know what you're looking for.

    Code (Text):
    a[k_] = (-1) k^3 + k^2 + 6;
    b[k_] = Sin[k];
    c[k_] = Cos[k];
    Plot[{a[k], b[k], c[k]}, {k, 0, 3}]
    Reduce[a[k] > 0 && b[k] > 0 && c[k] > 0, k] // FullSimplify
    So the result :
    c1 elem of Z OR c1 <= 0 OR 0 <k - 2 pi c1 < pi/2

    Means its true if (looking at the second two) if c1(some constant) is negative, so that

    k - 2 pi n < pi/2
    basically the first solution being
    0< k < pi/2 (which is the region thats positive where all three are positive).

    It's not perfect, but its correct.
     
  4. Apr 22, 2010 #3
    Hey there Hepth,

    this is what I have:

    A = 576 - 11819*K
    B = 15848*K
    C = C = 397119*K - 33.6*B/A

    and I need to find the range of K such that A,B, and C all remain greater than 0.
     
  5. Apr 22, 2010 #4

    Hepth

    User Avatar
    Gold Member

    A = 576 - 11819*k;
    B = 15848*k;
    CC = 397119*k - 33.6*B/A;
    Plot[{A, B, CC}, {k, 0, 0.04862}, PlotRange -> {0, 1000}]
    Reduce[A > 0 && B > 0 && CC > 0, k] // FullSimplify
    Reduce[A > y && B > y && CC > y && y > 0, y] // FullSimplify

    Gives for k : 0<k<0.0486216 (WHERE they're all positive)



    And for Y (bounded by this region):
    [tex]
    y>0\land \left((k>0\land y<15848. k\land k\leq 0.020819)\lor (k>0.020819\land 11819. k+y<576.\land k\leq 0.0486216)\lor \left(k>0.0486216\land [/tex]
    [tex]
    \landk<0.0486216\land y<\frac{k (2.34677\times 10^{10} k-1.14104\times 10^9)}{59095. k-2880.}\right)\right)
    [/tex]

    Which, while ugly, describes that range and domain bounded by the functions. I think you're only asking for the first one though.
     
    Last edited: Apr 22, 2010
  6. Apr 22, 2010 #5
    That is great Hepth! I think that will do the trick :smile: Thanks for your help.
     
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