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Math Help - Solve maximum problem without using calculus

  1. #1
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    Solve maximum problem without using calculus

    The problem is the following:

    max(cos(xy)+cos(xz)+cos(yz)), where

    a) x+y+z=180 degrees (or Pi);
    b) x,y,z are the angles of a triangle.

    Problem should be solved WITHOUT using calculus (derivatives etc.). One can use inequalities, properties of trigonometrical functions etc.

    Please help to solve this problem!!!
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  2. #2
    MHF Contributor alexmahone's Avatar
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    The maximum value of the cosine is 1.

    Each of the cosines should be 1 for that value. By symmetry, x=y=z=60^0 at that point.

    max(cos(xy)+cos (yz)+cos(yz))=cos (3600^o)+cos (3600^o)+cos (3600^o)=1+1+1=3
    Last edited by alexmahone; June 27th 2009 at 06:50 AM.
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  3. #3
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    CORRECT version of the problem

    Sorry...
    The CORRECT version is the following:

    max(cos(x)cos(y)+cos(x)cos(z)+cos(y)cos(z)), where

    a) x+y+z=180 degrees;
    b) x,y,z are the angles of a triangle.
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  4. #4
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    This is just by plugging numbers in, but I think that the angles of the triangle are still the same as alexmahone stated, ie. x=y=z=60^{\circ}.

    So
    max(\cos x \cos y + \cos x \cos z + \cos y \cos z)
    \begin{aligned}<br />
&= \cos 60^{\circ} \cos 60^{\circ} + \cos 60^{\circ} \cos 60^{\circ} + \cos 60^{\circ} \cos 60^{\circ} \\<br />
&= (0.5)(0.5) + (0.5)(0.5) + (0.5)(0.5) \\<br />
&= 0.75<br />
\end{aligned}


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