If xyz = (1-x)(1-y)(1-z) where 0 x, y, z 1 , then the minimum value of x(1-z) + y(1-x) + z(1-y) is:

Step-by-step solution

Let E = x(1-z) + y(1-x) + z(1-y) = (x+y+z) - (xy+yz+zx) . We are given xyz = 1 - (x+y+z) + (xy+yz+zx) - xyz . This implies 2xyz = 1 - ((x+y+z) - (xy+yz+zx)) = 1 - E . Therefore, E = 1 - 2xyz . To minimize E , we must maximize xyz . By AM-GM on (1-x), (1-y), (1-z) similar to the previous question, maximum xyz occurs when x=y=z . If x=y=z , then x^3 = (1-x)^3 x = 1-x x = 1 2 . So maximum xyz = 1 8 . Minimum E = 1 - 2( 1 8 ) = 1 - 1 4 = 3 4 .