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 ?
(In reply to
Whatever happened to that textbook...? by TomM)
Why partial differentials? Plain old Lagrange multipliers still work.
grad[P(a, b, c)] = L grad[g(a, b, c)]
{-a^-2, -2b^-2, -3c^-2} = L{1, 1, 1}
a^-2 = -L
2b^-2 = -L
3c^-2 = -L
a^2 = (b^2)/2 = (c^2)/3
b = a sqrt(2)
c = a sqrt(3)
a + a sqrt(2) + a sqrt(3) = 1
a (1 + sqrt(2) + sqrt(3)) = 1
a = 1/(1 + sqrt(2) + sqrt(3)) ~= 0.24118
b = a sqrt(2) ~= 0.34108
c = a sqrt(3) ~= 0.41774
P = 1/a + 2/b + 3/c = (1 + sqrt(2) + sqrt(3))^2 ~= 17.1915
As TomM mentioned, there is no maximum value for P.
re: Whatever happened to that textbook...?
