If -1 < x < 1, which of the following must be true?
I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2
Pick one:
A) I only
B) I and II only
C) II and III only
D) I and III only
E) I, II and III
I. x^3 &lt; x^2 --&gt; from -1 &lt; x &lt; 1 x must be a decimal so multiplying it by itself makes it a smaller value positive or negative. x^2 is always positive so x^3 being a small value always makes x^3<x^2 --="" 1="" <="" ii.="" is="" so="" statement="" this="" true.="" x="" x^5="" –=""> x(x^4+1) &lt; 1 --&gt; since 1(1+1)</x^2>=2 &gt; 1, so this statement is false.   III. x^4 &lt; x^2 --&gt; reduce by x^2 (we can safely do that since from -1 &lt; x &lt; 1 is follows that x^2&gt;0): x^2 &lt; 1. Again, as -1 &lt; x &lt; 1, then x^2 must be less than 1. Hence, this statement is also true.   Answer: I and III only.
<result>D</result>
If -1 < x < 1, which of the following must be true?
I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2