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avatar+1005 

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!

 Oct 25, 2022
 #2
avatar+118132 
+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

 Oct 25, 2022
 #3
avatar+1005 
+1

Thanks melody, I liked your reasoning.

Yes the answer was k = 1 for the optimal solution --- thanks, proyaop.

 Oct 25, 2022
 #4
avatar+118132 
0

You are very welcome.

It is not a proof though...

Melody  Oct 25, 2022

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