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# Diffrentiation

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Intern
Joined: 09 Aug 2018
Posts: 4

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09 Aug 2018, 07:14
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There are 4 positive integers whose sum is 4Q+3. S denotes the sum of the products of the four integers taken two at a time.. then what will be max/min value of S

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Manager
Joined: 02 Apr 2018
Posts: 50

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09 Aug 2018, 07:16
1
by taken two at a time, do you mean added?
Intern
Joined: 09 Aug 2018
Posts: 4

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09 Aug 2018, 07:37
Sum of the products of integers as : ab+ac+ad+bc+bd+cd

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Senior Manager
Joined: 18 Jun 2018
Posts: 252

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09 Aug 2018, 08:03
Srishti97 wrote:
Sum of the products of integers as : ab+ac+ad+bc+bd+cd

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$$(a+b+c+d)^2 = a^2+b^2+c^2+d^2 + 2(ab+ac+ad+bc+bd+cd)$$

$$ab+ac+ad+bc+bd+cd =\frac{(a+b+c+d)^2 - (a^2+b^2+c^2+d^2)}{2}$$

given $$a+b+c+d=4Q+3$$
$$a,b,c,d$$ are positive integers, their minimum value is 1
When $$(a^2+b^2+c^2+d^2)$$ is maximum, then $$ab+ac+ad+bc+bd+cd$$ will be minimum for a given $$a+b+c+d$$.
$$(a^2+b^2+c^2+d^2)$$ will be maximum when $$a=1,b=1,c=1$$ and $$d =4Q$$(any three of them can be 1 ,4th one have to be $$4Q$$)

Minimum value of $$ab+ac+ad+bc+bd+cd =1+1+4Q+1+4Q+4Q = 12Q + 3$$
Is this correct?
VP
Status: It's near - I can see.
Joined: 13 Apr 2013
Posts: 1354
Location: India
GMAT 1: 480 Q38 V22
GPA: 3.01
WE: Engineering (Consulting)

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09 Aug 2018, 08:17
Srishti97 wrote:
There are 4 positive integers whose sum is 4Q+3. S denotes the sum of the products of the four integers taken two at a time.. then what will be max/min value of S

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1. http://gmatclub.com/forum/11-rules-for- ... 33935.html

2. http://gmatclub.com/forum/rules-for-pos ... l#p1096628
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Re: Diffrentiation &nbs [#permalink] 09 Aug 2018, 08:17
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