Need help!!!! Minimum value problem.?

Question

Given x,y,z > 0 and xy+yz+zx <= 8/9
and S=x+y+z+(2/x)+(2/y)+(2/z)
Find the Minimum Value of S and (x,y,z) for that S

Answer 1

the answer appears to be 12.73 when x=y=z=0.54

my way of solving is:

since the variables are symmetric, the solution has to be of the form x=y=z=a

which then reduces the given conditions to x^2 <= 8/27 and S=3(x+2/x)

finding the minimum of S now is easy.... differentiate it to get x=sqrt(2) which can be proved to be the local minima

but this value will not satisfy the given condition...

since S is monotonically increasing function, we can take x=sqr(8/27) as the minimum value

Answer 2

A hint

First do the unconstrained problem using derivatives. as x,y,z -> 0 or x,y,z -> infty then S -> infty and this is a continuous infinitely differentiable function... so go ahead take derivatives and set to 0 (i.e., find critical points)

Doing this you will find that (sqrt(2),sqrt(2),sqrt(2)) is the only critical point. (i.e., solution to 0 = S_x = 1 - 2/x^2)

It turns out that this is a global minimum (2nd deriv test) but does not meet your constraint. Since 6 > 8/9

So then it must be that the minimum occurs on the boundary...

Then you have a constraint that you can use lagranian multipliers on... i.e., xy+yz+zx -8/9 = 0

So S = x + y + z + 2/x + 2/y + 2/z - L * (xy + yz + zy - 8/9)