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 .