# Thread: Solve maximum problem without using calculus

1. ## 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.

2. The maximum value of the cosine is 1.

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

$\displaystyle max(cos(xy)+cos (yz)+cos(yz))=cos (3600^o)+cos (3600^o)+cos (3600^o)=1+1+1=3$

3. ## 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.

4. This is just by plugging numbers in, but I think that the angles of the triangle are still the same as alexmahone stated, ie. $\displaystyle x=y=z=60^{\circ}$.

So
$\displaystyle max(\cos x \cos y + \cos x \cos z + \cos y \cos z)$
\displaystyle \begin{aligned} &= \cos 60^{\circ} \cos 60^{\circ} + \cos 60^{\circ} \cos 60^{\circ} + \cos 60^{\circ} \cos 60^{\circ} \\ &= (0.5)(0.5) + (0.5)(0.5) + (0.5)(0.5) \\ &= 0.75 \end{aligned}

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