Given:

   a,b,c >0 and a+b+c=1 ;
   P=(1/a)+(2/b)+(3/c);

1)Find the minimum value of P;
2)Does P have maximum value ?

P*1 = (a+b+c)*(1/a+2/b+3/c);
P*1=6+2a/b + 3a/c + b/a + 3b/c + c/a
+ 2c/b
P=6+(2a/b+b/a)+(3a/c+c/a)+(3b/c+2c/b)
Use Cauchy's inequality for each :
we have :
P >= 6+ 2*sqrt(2)+2*sqrt(3)+2*sqrt(6)

"=" a+b+c=1 and b=a*sqrt(2)
c=a*sqrt(3)
===>we can solve for a,b,c
No maximum value.

Posted by vohonam on 2002-06-21 11:58:55