Given that x, y, and z are variables in the interval [0, 1], what is the maximum value of k = x + y(1 - x) + z(1 - x)(1 - y)?

Please help me find k (also [0, 1] means all real numbers between 0 and 1 inclusive)! Any help appreciated -- thanks guys!

proyaop Oct 25, 2022

#2**+2 **

\(k = x + y(1 - x) + z(1 - x)(1 - y)\\ k = x + y-xy + z(1-x-y+xy)\\ k = (x + y-xy) - z(x+y-xy)+z\\ k = (x + y-xy) (1- z)+z\\\)

The maximum of x+y-xy is 1 when x=1 and y=1

You can prove that by considering the case where x=1-t and y=1-m where t and m are (0,1]

If z=0 then we have 1*1+0 = 1

So I get a max of 1 when x=1,y=1 and z=0

There is also be a max of 1 when x=0, y=0 and z =1

I do not think that k can be bigger than 1

Melody Oct 25, 2022