Solve maximum problem without using calculus

**Krabathor** · June 27th 2009, 03:11 AM

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!!!

**alexmahone** · June 27th 2009, 06:06 AM

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.

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

**Krabathor** · June 28th 2009, 01:38 AM

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.

**yeongil** · June 28th 2009, 05:02 AM

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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