Bunuel wrote:

If f is a fraction between -1 and 1, which of the following must be true?

I. \(f^6 - f^7 < f^4 - f^5\)

II. \(f^6 + f^7 < f^4 + f^5\)

III. \(f^6 < f^4\)

A. I only

B. II only

C. III only

D. I and III only

E. I, II and III

For any fraction between -1 and 1, we know for sure that higher powers means lower absolute values. And even powers ensures positive values whereas negative powers could be positive or negative depending on the fraction.

Keeping this in mind.

I. \(f^6 - f^7 < f^4 - f^5\)

Re-arranging the terms...

\(f^5 - f^7 < f^4 - f^6\)

The RHS is a positive value as f^4 is greater than f^6 and LHS is either a negative value hence the inequality holds when f is negative.

\(f^5( 1-f^2) < f^4( 1-f^2)\) since \(1-f^2\) is positive we can cancel it out and hence the inequality holds.

II. \(f^6 + f^7 < f^4 + f^5\)

Re-arrange..

\(f^7 - f^5< f^4 - f^6\)

The RHS is positive where as LHS is negative when f is positive, hence inequality holds.

when f is negative,

\(f^5 ( f^2 -1) < f^4 ( 1- f^2)\)

\(-f^5 (1-f^2) < f^4 ( 1- f^2)\)

This holds.

III. Always holds as increasing powers of f are smaller.

Hence Option (E) is our choice.

Best,

Gladi

_________________

Regards,

Gladi

“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back)