Powerprep

[Cybermusings]

If x is positive which of the following could be the correct ordering of 1/x, x^2, 2x

1) x^2 < 2x <1/x

2) x^2 < 1/x < 2x

3) 2x < x^2 < 1/x

1) None
2) I alone
3) III alone
4) I and II
5) I, II and III

[Sadowski]

I'm going with B (I alone) based on 0<x<1 (eg x=.5, .25<1<2)

The others don't work out regardless of whether x is less than or greater than 1.

[jayhawk2001]

If x is positive which of the following could be the correct ordering of 1/x, x^2, 2x

1) x^2 < 2x <1/x

2) x^2 < 1/x < 2x

3) 2x < x^2 < 1/x

1) None
2) I alone
3) III alone
4) I and II
5) I, II and III

With these kinds of questions, it is always useful to find the boundaries

We have 3 equations giving us 3 boundaries --

1/x = x^2 implies x = 1
1/x = 2x implies x = 0.7 approx
2x = x^2 implies x = 2

Our 3 boundaries are 0.7, 1 and 2

Take 1 sample for each range

For x<0.7, try x = 0.5
We have x^2 < 2x < 1/x ... bingo, you have (I) satisfied

For x>0.7 and <1, try x = 0.8
we have x^2 < 1/x < 2x ... bingo, you have (II) satisfied

It is easy to see that for all values x > 1, 1/x will give you the
least of the 3. So, (III) can never be true.

So, choose D (i.e. I and II alone)

[Sadowski]

I stand corrected.

Jayhawk -

Good call. Nice strategy.

Learn something new every day.