Which of the following inequalities are satisfied for all real numbers a, b, c, x, y, z that satisfy the conditions x < a, y < b, and z < c?
- xy + yz + zx < ab + bc + ca
- x2 + y2 + z2 < a + b + c
- xyz < abc
- None is satisfied
- I only
- II only
- III only
- All are satisfied
We search vigorously for counterexamples in which x < a, y < b, and z < c
-
xy + yz + zx < ab + bc + ca
Let x = y = z = -1 and a = b = c = 0 and I is false. -
x2 + y2 + z2 < a + b + c
We can use the same numbers to destroy II. -
xyz < abc
Let x = y = -1 and z = 1, and a = b = 0 and c = 2. That wrecks III.
So the answer is (a): None of the inequalities is satisfied.