If x is positive, which of the following could be the correct ordering of 1/x , 2x and x^2?
1. x^2 < 2x < 1/x
2. x^2 < 1/x < 2x
3. 2x < x^2 <1/x
I got that 1 is possible but from 2 and 3 another one is also possible according to the answer. Can someone please explain
x=1/2 I is possible (every body agrees with this)
x^2 < 1/x < 2x
--> x^2-1/x<0 and 1/x-2x<0
x^3-1<0 and 2x*x-1>0 (x is positive)
(x-1)*(x2+1)<0 and x*x>1/2
(because x is postiive)
--> x<1 and x> 1/sqrt(2)
So its possible for any value > 1/sqrt(2) and <1
I and II possible.
2x < x^2 <1/x
x^2-2x>0 and x^2-1/x<0
x*(x-2)>0 and x3-1<0
you know that x>0 (positive)
x-2>0 and x-1<0
x>2 and x<1 not possible at all.
only I and II possible.
Your attitude determines your altitude
Smiling wins more friends than frowning