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

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

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New post 01 Nov 2018, 01:26
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

60% (02:03) correct 40% (02:08) wrong based on 56 sessions

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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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New post 01 Nov 2018, 01:41
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,
Gladi
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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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New post 01 Nov 2018, 04:13
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.

Answer choice E
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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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New post 04 Nov 2018, 02:34
Is 0/1 not considered a fraction?
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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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New post 04 Nov 2018, 02:45
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   [#permalink] 04 Nov 2018, 02:45
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