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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim

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Intern  B
Joined: 29 Aug 2010
Posts: 9
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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5 00:00

Difficulty:   55% (hard)

Question Stats: 59% (01:44) correct 41% (02:00) wrong based on 198 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 29 Oct 2017, 23:19.
Last edited by Bunuel on 29 Oct 2017, 23:43, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Manager  P
Joined: 15 Feb 2017
Posts: 70
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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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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##### General Discussion
Intern  B
Joined: 07 Oct 2018
Posts: 46
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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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?
Intern  B
Joined: 18 Aug 2017
Posts: 4
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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i got 1/2 as answer.. can someone explain?
Manager  B
Joined: 24 Jun 2017
Posts: 113
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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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
Intern  Joined: 08 May 2020
Posts: 1
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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Shreshtha55 wrote:
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?

The question is asking the minimum value of f(g(x) and what you are telling is the minimum value of x (i.e 1 ) as an answer , when you will use x =1 in the given equation you will get the output as 2 , that's the answer for the question, other values will give output greater than 2 . Option A will give an output as 1 but , the condition states that x has to be greater than 0.
If the question would have asked the minimum value of x , ans would have been Option C, but here , question states what is minimum value of f(g(x) , which is 2. So Option E

Hope its clear.

L. If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim   [#permalink] 13 May 2020, 07:00

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