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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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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"
Behavior of 'X' in different regions.png [ 31.35 KiB | Viewed 1767 times ]
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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