Question

If x,y,z are non negative integers so that x + y + z = 12, then the

maximum value of xyz + xy + yz + zx is

• A. 115
• B. 112
• C. 114/3
• D. 111

Answer 1

The correct option is B

112

1.We have(x + 1) (y + 1) (z + 1) = xyz + xy + yz + zx + x + y + z + 1 = xyz + xy + yz + zx + 13;
And (x + 1) (y + 1) (z + 1) = 15. Using AM.GM inequality we get xyz + xy + yz + zx + 13 ≤ =125, xyz + xy + yz + zx ≤ 112. Equality holds if x + 1 = y + 1 = z + 1 ; i.e., x = y = z = 4.

So the desired maximum is 112. The answer is B.