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

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Math Expert
Joined: 02 Sep 2009
Posts: 50730
If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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01 Nov 2018, 00:26
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Difficulty:

45% (medium)

Question Stats:

63% (01:30) correct 37% (01:51) wrong based on 38 sessions

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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

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Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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01 Nov 2018, 00:41
1
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,
Director
Joined: 19 Oct 2013
Posts: 503
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Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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01 Nov 2018, 03:13
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

f^6 ( 1 - f) < f^4 (1 - f) as the power gets larger the value comes smaller. so this holds true regardless of the sign. The (1-f) cancels and we have f^6 < f^4 similar to III

f^6 (1+f) < f^4 ( 1+f) this also holds true in a similar way to the above. The (1+f) cancels and we have f^6 < f^4 similar to III

f^4 ( f^2 - 1) < 0 whatever is inside the bracket is negative so +ve * -ve < 0 is true.

Manager
Joined: 28 Jun 2018
Posts: 52
Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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04 Nov 2018, 01:34
Is 0/1 not considered a fraction?
Director
Joined: 19 Oct 2013
Posts: 503
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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04 Nov 2018, 01:45
hibobotamuss 0/any value other than zero = 0

Thus, it is an integer

This comes from the relationship that 0 is a multiple of every number.

0*1 = 0 etc..

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Re: If f is a fraction between -1 and 1, which of the following must be tr &nbs [#permalink] 04 Nov 2018, 01:45
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# If f is a fraction between -1 and 1, which of the following must be tr

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