Must be true question

rahulvsd wrote:If a^5 ≤ a, which of the following must be true?
I –1 ≤ a ≤ 0
II a=0
III 0 ≤ a ≤ 1


A. None of the above
B. I only
C. II only
D. III only
E. I and III only
[spoiler]
OA: A. Source: MasterGMAT[/spoiler]

Lets manipulate the given inequality to know more about 'a'.
a^5 ≤ a
a^5 - a ≤ 0
a(a - 1)(a + 1)(a^2 + 1) ≤ 0
Using the Critical Points' method,
a ≤ -1 or 0 ≤ a ≤ 1

Since it is a 'must be true' question, we now have to see which of the three statements cover the entire range of values of 'a'. If even one solution of 'a' is not included in a statement, we will have to rule it out.

(I) is ruled out because it is not even a solution of 'a'.
(II) is ruled out because a = 0 is just one solution in the huge range. We'll be missing out many other numbers if we accept this.
(III) is ruled out because if we accept it we will be leaving out a ≤ - 1.

[spoiler](A)[/spoiler] is correct.

P.S.: Critical Points' method is explained in detail here:
https://www.beatthegmat.com/critical-poi ... tml#465861

More on 'must be true' questions here:
https://www.beatthegmat.com/mba/2011/10/ ... -questions