If a, B, C, x, y, Z, > 0, X * 2 + y * 2 + Z * 2 = 1, find the minimum value of F (x, y, z) = A / x + B / y + C / Z

If a, B, C, x, y, Z, > 0, X * 2 + y * 2 + Z * 2 = 1, find the minimum value of F (x, y, z) = A / x + B / y + C / Z

Let f (x, y, z) = 0. λ + F (x, y, z) = λ (x * 2 + y * 2 + Z * 2-1) + F (x, y, z) = λ (x * 2 + y * 2 + Z * 2-1) + A / x + B / y + C / Z,
be
Partial f / partial λ = x * 2 + y * 2 + Z * 2-1 = 0; ①
Partial f / partial x = 2 λ · x-a / x ^ 2; 2
Partial f / partial y = 2 λ · y-b / y ^ 2; ③
Partial f / partial z = 2 λ · z-c / Z ^ 2; 4
Let (2) (3) (4) = 0, then
x=[a/(2λ)]^(1/3)
y=[b/(2λ)]^(1/3)
z=[c/(2λ)]^(1/3);
With the solution of (1), λ = (1 / 2) · [a ^ (2 / 3) + B ^ (2 / 3) + C ^ (2 / 3)]
Then you can get it
x=a^(1/3)·[a^(2/3) +b^(2/3) +c^(2/3) ]^(-1/2)
y=b^(1/3)·[a^(2/3) +b^(2/3) +c^(2/3) ]^(-1/2)
z=c^(1/3)·[a^(2/3) +b^(2/3) +c^(2/3) ]^(-1/2)
That is, when x, y, Z take the above values respectively, f (x, y, z) gets the minimum value, which is
[a^(2/3) +b^(2/3) +c^(2/3) ]^(3/2)