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 ?

The maxima would occur when one or two of a, b, and c is 0, except that P is undefined at these points. As any one of the independant variables becomes arbitrarily close to zero, P becomes arbitrarily large.

The minimum is little more involved. If the numerators of the fractions in the problem were all equal, it would occur when a = b = c = 1/3, but they are different. It occurs when b = 2a and c = 3a. Since 1 = a + b + c = a + 2a + 3a = 6a, we get a = 1/6, giving us b = 1/3 and c = 1/2. At this point we get

P = [1/(1/6)] + [2/(1/3)] + [3/(1/2)]
P = [6] + [2 ⋅ 3] + [3 ⋅ 2]
P = 6 + 6 + 6 = 18

Posted by TomM on 2002-06-20 20:09:57