What is max[$XY + YZ + ZX$] if $X^2 + Y^2 + Z^2 =1$?

Question

For real $X,Y,Z$, how to find the maximum of $XY+YZ+ZX$ subjected to condition $X^2 + Y^2 + Z^2 =1$? I am aware of the fact that for a single variable function $f(x)$, one would simply find the critical points and apply the second derivative test. But how can one handle this multiparameter problem?

Answer 1

Who needs derivatives?

$1-(xy+xz+yz)=(x^2+y^2+z^2)-(xy+xz+yz)$

$=(\frac12x^2-xy+\frac12y^2)+(\frac12x^2-xz+\frac12z^2)+(\frac12y^2-yz+\frac12z^2)$

$=\frac12(x-y)^2+\frac12(x-z)^2+\frac12(y-z)^2\ge0.$

Thus $xy+xz+yz\le1$, and since equality is met by rendering $x=y=z=\sqrt{\frac13}$, this upper bound of $1$ represents the true maximum.

This problem is homogeneous, so for any nonnegative $c$ we find that if $x^2+y^2+z^2=c$ then the maximum of $xy+xz+yz$ will also be $c$.