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If x > 1, then it is always true that x < x^2 < x^3.

If 0 < x < 1, then it is always true that x^3 < x^2 < x.

From the above, you can see that neither statement is sufficient alone, since in each case, x can be positive. Notice from the above that if x is positive, x^2 is never the largest of the three expressions x, x^2 and x^3. Since Statement 1 guarantees that x^3 is not the largest of the three expressions, and Statement 2 guarantees that x is not the largest of the three expressions, then using both statements, the only possibility is that x^2 is the largest of the three expressions. Since that can't happen when x is positive, x must be negative, and the answer is C.
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Can someone please explain me what "at least one of" means in the statements? I really have no idea how to convert this sentence into an equation.

At least one of x and x^2 is greater than x^3 means that x>x^3 OR x^2>x^3 OR x>x^3 and x^2>x^3 (so, x is greater than x^3 or x^2 is greater than x^3 or both are greater than x^3).
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I documented the behavior of a variable in different regions. Uploading its image as I think it would be helpful.

Although, I believe, memorizing how every power of 'x' behaves in each region would be pointless, noticing the patterns such as the ones mentioned below would be useful. -- The behavior of odd powers of 'x' in the region " -1 < x < 0" is exactly same as that of even powers in the region "x < -1" -- The behavior of even powers of 'x' in the region " -1 < x < 0" is exactly same as that of odd powers in the region "x < -1"

Attachments

Behavior of 'X' in different regions.png [ 31.35 KiB | Viewed 1772 times ]

You can see how x, x^2 and x^3 behaves from the graph attached.. We can then answer the qn accordingly

Is x negative?

(1) At least one of x and x^2 is greater than x^3. (2) At least one of x^2 and x^3 is greater than x.

Lets define the regions : A <-1 , -1< B <0 , 0<C<1 , 1<D;

Blue line - x ,Red line - x^2 , Green line - x^3

(1) So the region can be either of A , B or C.. It can be either positive or negative (2) So the region can be either of A , B or D .. It can be either positive or negative

Each is insufficient. Now combine both (1) and (2)...

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