Max value

Dec 2015
975
128
Earth
Find the maximal value of function \(\displaystyle F(x,y,z)=\sin(xy)\cdot \sin(yz)\cdot \sin(xz)\).
 

romsek

Math Team
Sep 2015
2,891
1,615
USA
We know that each term has a maximum value of 1 so the product has a max value of 1.

We just need to demonstrate that the product does in fact reach 1.

Thus we need to solve for

$x y = x z = y z = \dfrac{\pi}{2}$

certainly a solution is $x = y = z = \sqrt{\dfrac \pi 2}$

and we only need one