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a, b, c and d are four positive real numbers such that abcd=

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a, b, c and d are four positive real numbers such that abcd= [#permalink] New post 08 Nov 2009, 12:53
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a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)?

A. 4
B. 1
C. 16
D. 18
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Jul 2013, 13:59, edited 1 time in total.
Added the OA
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Re: Algebra....aproach? [#permalink] New post 08 Nov 2009, 14:28
Expert's post
papillon86 wrote:
a,b,c and d are four +ve real nubers such that abcd=1, what is the minimum value of
(1+a)(1+b)(1+c)(1+d)?

a) 4
b) 1
c) 16
d) 18


what is the best way to solve such question where in we need to calculate the min or max values?


Think there is no catch in this question. As the numbers are positive and their product is 1: either 2,3, or all 4 numbers are reciprocals and rest is 1 OR all numbers are equal to 1.

Minimum value will be when a=b=c=d=1, hence (1+a)(1+b)(1+c)(1+d)=16. (You can try reciprocals to see that the product will be greater)

Answer:C

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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] New post 14 Jul 2014, 13:06
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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] New post 14 Jul 2014, 21:40
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papillon86 wrote:
a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)?

A. 4
B. 1
C. 16
D. 18


As Bunuel said, abcd = 1 implies that either the numbers are equal to 1 or there are pairs of reciprocals e.g. (1, 1, 1, 1) or (1, 1, 2, 1/2) or (3, 1/3, 4, 1/4) etc.

If a and b are 1 and 1,
(1+a)(1+b) = (1+1)(1+1) = 4

If a and b are 2 and 1/2,
(1+a)(1+b) = (1+2)(1+1/2) = 9/2 = 4.5

If a and b are 3 and 1/3,
(1+a)(1+b) = (1+3)(1+1/3) = 16/3 = 5.3

As you keep taking higher reciprocals, the value of (1+a)(1+b) keeps increasing.

So taking reciprocals is a bad idea and all numbers must be 1 giving us the minimum value of 16.

Anyway, in any minimum-maximum question, it is a good idea to check on equality. Often, the point of equality is a transition point.

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Re: a, b, c and d are four positive real numbers such that abcd=   [#permalink] 14 Jul 2014, 21:40
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