We know that x is positive and we see that we're comparing 2x to the reciprocal of x to x squared.

Let's think about positive numbers that behave differently:

fractions
1
2 (since 2x = 2^2)
numbers bigger than 2

When we test the answer choices, those are the different types of numbers we should choose.

As Kaplan recommends on roman numeral questions, let's start by looking at the answer choices to see if any numerals occur more frequently than others:

(a) none
(b) I only
(c) III only
(d) I and II only
(e) I, II and III

I appears most often, so let's start there.

I x^2 < 2x < 1/x

If we pick x = 1/2, we get:

1/4 < 1 < 2,

which IS true. Therefore, I COULD be true. Elminate (a) and (c).

II occurs next most often, let's look at that one:

II x^2 < 1/x < 2x

We know that 1 won't work (since 1^2 = 1/1). For any number greater than 1, x^2 will be greater than 1/x. So, our only hope is find a different fraction.

Let's try a bigger fraction. If we pick x=9/10, we get:

81/100 < 10/9 < 18/10,

which is true! So, II COULD also be true and should be included in our answer.

Sadly, we can only eliminate (b). We have two choices left, so we have to test III.

III 2x < x^2 < 1/x

Again, we can quickly eliminate 1 and numbers bigger than 1. All that's left are fractions, which we can also eliminate, since:

2 * fraction is always MORE THAN fraction * fraction (which is x^2).

So, there are no positive values that will make III work. Elminate (e) and choose (d) I and II only.