If x is positive, which of the following could be the correct ordering of 1/x, 2x, and x²?

I. x² < 2x < 1/x
II. x² < 1/x < 2x
III. 2x < x² < 1/x

a. None
b. I
c. III
d. I and II
e. I, II, and III

Determine the CRITICAL POINTS by setting the expressions equal to each other:

1/x = 2x
2x² = 1
x² = 1/2
x = √(1/2) = 1/√2 ≈ 1/1.4 ≈ 10/14 ≈ 5/7.

1/x = x²
x^3 = 1
x = 1.

2x = x²
x=2
(We can divide by x because x>0.)

The critical points are x=5/7, x=1, x=2.
These critical points indicate where two of the expressions are EQUAL.
Thus, to the left and right of each critical point, the value of one expression must be GREATER than the value of another.

To determine which of I, II and II could be true, plug in values to the left and right of each critical point.
Start with the range that many test-takers will fail to consider: 5/7 < x < 1.

5/7 < x < 1:
If x = 3/4, then:
1/x = 4/3.
x² = 9/16.
2x = 3/2.
Since x² < 1/x < 2x, we know that II could be true.
Eliminate A, B and C.

In statement III, 2x<x², which implies that 2<x.
But if x>2, then 1/x cannot be the greatest of the three values.
Thus, III is not possible.
Eliminate E.

The correct answer is D.

Greeting Mr. Hunt
Your explanation is great.
I was stuck at II.
Now it is clear.
Thank you very much.
Cheer.

If we follow this:

x | 1/x | 2x | x^2
-------------------
1 | 1 | 2 | 1
2 | .5 | 4 | 4
3 | .33 | 6 | 9

we can see no pattern is visible between 2x , 1/x and x^2. So none could be the answer .
Whats wrong in this approach ?

ygdrasil24 wrote:If we follow this:

x | 1/x | 2x | x^2
-------------------
1 | 1 | 2 | 1
2 | .5 | 4 | 4
3 | .33 | 6 | 9

we can see no pattern is visible between 2x , 1/x and x^2. So none could be the answer .
Whats wrong in this approach ?

The question stem does not ask which of the three statements MUST be true.
Rather, it asks which of the three statements COULD be true.

Since x=5/7 yields x² < 1/x < 2x, statement II COULD be true.
Thus, the correct answer choice must include statement II.
Eliminate A, B and C.

Since no positive value of x will yield 2x < x² < 1/x, statement III could NEVER be true.
Thus, the correct answer choice cannot include statement III.
Eliminate E.

The correct answer is D.

Okay...so i just undermined what is being asked " could be" than " should be".
In real test,its a sin to miss out such quesns because of such mistakes.
Thanks indeed!