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If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim

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If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim  [#permalink]

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New post Updated on: 30 Oct 2017, 00:43
2
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

58% (01:49) correct 42% (01:54) wrong based on 158 sessions

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If \(f(x)=\frac{1}{x}\) and \(g(x) = \frac{x}{(x^2+1)}\), for all \(x > 0\), what is the minimum value of \(f(g(x))\)?

(A) 0

(B) \(\frac{1}{2}\)

(C) 1

(D)\(\frac{3}{2}\)

(E) 2

Originally posted by shibbirahamed on 30 Oct 2017, 00:19.
Last edited by Bunuel on 30 Oct 2017, 00:43, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim  [#permalink]

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New post 30 Oct 2017, 03:11
8
IMO Option E
f(g(x)) = \(\frac{x^{2} +1}{x}\)
=x + \(\frac{1}{x}\)
=(\(\sqrt{x}\) -\(\frac{{1}}{{\sqrt{x}}}\))^2+2
The value squared is zero. So the minimum value of the expression is 2.
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Re: If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim  [#permalink]

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New post 08 Nov 2017, 14:20
i got 1/2 as answer.. can someone explain?
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Re: If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim  [#permalink]

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New post 08 Nov 2017, 16:58
it's not a gmat type question first of all
second the solution of user siva is mathematically wrong, you cannot divide by 0 and you can't have 0 under the root. I'm not sure if it can serve as shortcut or it's just a coincidence. There is no way in the real number system that root of x can become 0, but you can assume that there is some x when (√x -1/√x)^2 can take 0
Normally the question should say that x>0 and integer.


answer is indeed 2 because if you think about it x + 1/x
and x > 0 (it can be a decimal as well as per problem description)
basically when x is minimum (0,00000000.......1) so lim x -> +0 then x + 1/x has no solution as 1/0 is undefined.

And if you can just pick up the next integer 1 then x + 1/x so 1+ 1 = 2
check one more 2 -> 2 + 1/2 = 2,5
check x=1/2 -> 1/2 + 1/(1/2) = still 2,5


looking forward to checking other solutions
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Re: If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim  [#permalink]

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New post 09 Aug 2019, 07:12
1
Can someone explain with process how can e) be the ans. when I substituted all the values in x+1/x. I got ans C). Even after using derivation method I got C) as answer not E). What am I missing here?
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Re: If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim   [#permalink] 09 Aug 2019, 07:12
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