If a and b are both positive numbers,and

a>b,which of the following could be true?
I. ab is greater than either a or b.
II. ab is greater than b but less than a.
III. ab is less than either a or b.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I,II,and III

Assuming that a and b are both ≥ 1, A.

To determine which of the statements could be true, let's consider the given information: a>b. This tells us that a is greater than b.

Now, let's analyze each statement:

I. ab is greater than either a or b.
To check if this statement could be true, we compare ab with a and b.
- If a is greater than b, then ab will also be greater than b, since a*b > b.
- However, ab may or may not be greater than a. It depends on the value of b. If b is a fraction or less than 1, then ab would be less than a. But if b is greater than 1, then ab would be greater than a.
Therefore, statement I could be true.

II. ab is greater than b but less than a.
To check if this statement could be true, we compare ab with b and a.
- If a is greater than b, then ab will be greater than b, as mentioned earlier.
- However, ab cannot be simultaneously less than a and greater than b because a is greater than b. Therefore, statement II cannot be true.

III. ab is less than either a or b.
To check if this statement could be true, we compare ab with a and b.
- If a is greater than b, then ab will be greater than b, but less than a.
- Therefore, statement III could be true, as ab is less than a but greater than b.

Based on our analysis, the statements that could be true are I and III. Therefore, the answer is (D) I and III only.