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
An alternate approach is to 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.
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 for 1/x to be the greatest value, x must be a fraction.
Since x cannot at the same time be both a value greater than 2 and a positive fraction less than 1, III is not possible.
Eliminate E.
The correct answer is
D.
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