Suppose x; y, and z are positive numbers that sum to 10. What is the largest possible value of xy + xz + yz?

Question

(a) First, suppose that z is a �fixed parameter. Then we have to �find nonnegative numbers x and y (depending on the �fixed value z such that x+y = 10-􀀀z), such that xy + xz + yz is maximized.

(b) To fi�nd the global maximum, plug in the values for x and y obtained in (a) in
order to rewrite xy + xz + yz as a function of a single variable. Then maximize
this function.

Answer 1

since x+y+z = 10, we want the max of

F(x,y) = xy + (x+y)(10-x-y)
Fx = -2x-y+10
Fy = -x-2y+10

Fx = Fy = 0 at (10/3,10/3)
So, max of F(x,y) = 100/3
That is, when x,y,z are all equal, at 10/3.

Answer 2

Using common sense:

for any value xy, where x + y is a constant,
the largest value xy is obtained when xy is a perfect square,
So the largest value of xy+xz+zy is obtained when all three are the same value
but 10 cannot be split into 3 equal parts
so the best we can do is split them into
3,3, and 4
so largest sum = 3x3 + 3x4 + 3x4 = 33

confirmation: - a little program
5 big = 0
10 for x = 1 to 10
20 for y = 1 to 10
30 for z = 1 to 10
40 sum = x*y+x*z+y*z
45 if x+y+z > 10 then 60
50 if sum > big then big = sum : print x,y,z,sum
60 next z
70 next y
80 next x
>run
1 1 1 3
1 1 2 5
1 1 3 7
1 1 4 9
1 1 5 11
1 1 6 13
1 1 7 15
1 1 8 17
1 2 6 20
1 2 7 23
1 3 6 27
1 4 5 29
2 3 5 31
2 4 4 32
3 3 4 33 <--- sum
>