2006 SMT/Algebra Problems/Problem 2
Contents
[hide]Problem
Find the minimum value of for real numbers
.
Solution
Solution 1
Notice that . This achieves a minimum value when all of the squares are
, that is, when
and
. Solving, we find that
and
satisfy this, and so the minimum value is
.
Solution 2
Let . The minimum value occurs when
. Taking these partial derivatives, we have
From the first and third equations, we have and
. Plugging these into the second equation and solving, we find that
. From this we get
and
. Therefore, the minimum value is
.