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 answer choices are possible, plug in values to the left and right of each critical point.
We should start with the range most likely to be IGNORED by the average test-taker: 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 III, the largest value listed is 1/x.
For 1/x to be the largest value, x would have to be a fraction.
But if x is a fraction, then it is not possible that 2x<x², since doubling a fraction INCREASES its value, while squaring a fraction results in a SMALLER value.
Since III is not possible, eliminate E.
The correct answer is
D.
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