We have :

      x^2+xy+y^2=3 and
      y^2+yz+z^2=16
      A=xy+yz+zx

Find the maximum value of A. Find x, y and z when A=max value.

(Remember the category)

We use 2 inequalities :
1)Cauchy's inequality :
a + b >= 2*square root of (a*b) ; (a,b>0)
or a common Chauchy's inequality :
((a1+a2+a3+.....+a(n))/n)^n >= a1*a2*...*a(n)
2)Chebyshev's inequality :
absolute value of (a*x+b*y) <= square root of ((a^2+b^2)*(x^2+y^2))
"=" when a/x=b/y
Those inequalities are easy to prove.
From these, I hope that you can solve this problem.
"=" when a1=a2=...=a(n)

Posted by vohonam on 2002-06-19 17:27:00