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If x is positive, what is the least possible value of x/2 + 2/x?

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Bunuel
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If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   18 Mar 2020, 02:24
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If x is positive, what is the least possible value of \(\frac{x}{2} + \frac{2}{x}\)?

(A) 1/2
(B) 1
(C) 2
(D) 3
(E) 4


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C
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nick1816
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   18 Mar 2020, 03:20
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AM ≥ GM

\((\frac{x}{2} + \frac{2}{x})/2 ≥ \sqrt{\frac{x}{2}*\frac{2}{x}}\)

\((\frac{x}{2} + \frac{2}{x})/2 ≥ 1\)

\((\frac{x}{2} + \frac{2}{x}) ≥ 2\)

C

Bunuel wrote:
If x is positive, what is the least possible value of \(\frac{x}{2} + \frac{2}{x}\)?

(A) 1/2
(B) 1
(C) 2
(D) 3
(E) 4


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ArunSharma12
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   18 Mar 2020, 02:39
[quote="Bunuel"]If x is positive, what is the least possible value of \(\frac{x}{2} + \frac{2}{x}\)?

(A) 1/2
(B) 1
(C) 2
(D) 3
(E) 4[\quote]

it'll increase for lower values of x
for x = 1; \(\frac{x}{2} + \frac{2}{x}\) = 2.5
for x = 2; \(\frac{x}{2} + \frac{2}{x}\) = 2
for x = 3; \(\frac{x}{2} + \frac{2}{x}\) = 2.16
for x = 4; \(\frac{x}{2} + \frac{2}{x}\) = 2.5
and it'll increase further for higher values of x

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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   18 Mar 2020, 04:31
Bunuel wrote:
If x is positive, what is the least possible value of \(\frac{x}{2} + \frac{2}{x}\)?

(A) 1/2
(B) 1
(C) 2
(D) 3
(E) 4


Are You Up For the Challenge: 700 Level Questions


\(\frac{x}{2} + \frac{2}{x} - 2*\sqrt{x/2}\sqrt{2/x} = (\sqrt{x/2}- \sqrt{2/x})^2 >0\)
\(\frac{x}{2} + \frac{2}{x} > 2\)

IMO C
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   09 Apr 2020, 01:12
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nick1816 wrote:
AM ≥ GM

\((\frac{x}{2} + \frac{2}{x})/2 ≥ \sqrt{\frac{x}{2}*\frac{2}{x}}\)

\((\frac{x}{2} + \frac{2}{x})/2 ≥ 1\)

\((\frac{x}{2} + \frac{2}{x}) ≥ 2\)

C

Bunuel wrote:
If x is positive, what is the least possible value of \(\frac{x}{2} + \frac{2}{x}\)?

(A) 1/2
(B) 1
(C) 2
(D) 3
(E) 4


Are You Up For the Challenge: 700 Level Questions


Hi, could you please elaborate on your approach? Is there a definite way to solve such min/max problems??
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nick1816
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   09 Apr 2020, 02:53
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Assume x and y are 2 positive numbers

\((x-y)^2 ≥ 0\)

\((x+y)^2 - 4xy ≥ 0\)

\((x+y)^2 ≥ 4xy\)

\(\frac{(x+y)^2}{4} ≥ xy\)

\(\frac{x+y}{2} ≥ \sqrt{xy}\), where x,y>0

arithmetic mean ≥ geometric mean



Since x is positive in the question, 2/x and x/2 >0. You can use the above inequality.



Kritisood wrote:
nick1816 wrote:
AM ≥ GM

\((\frac{x}{2} + \frac{2}{x})/2 ≥ \sqrt{\frac{x}{2}*\frac{2}{x}}\)

\((\frac{x}{2} + \frac{2}{x})/2 ≥ 1\)

\((\frac{x}{2} + \frac{2}{x}) ≥ 2\)

C

Bunuel wrote:
If x is positive, what is the least possible value of \(\frac{x}{2} + \frac{2}{x}\)?

(A) 1/2
(B) 1
(C) 2
(D) 3
(E) 4


Are You Up For the Challenge: 700 Level Questions


Hi, could you please elaborate on your approach? Is there a definite way to solve such min/max problems??
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bellbell
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   14 Sep 2022, 02:09
Another approach:
Set x/2 + 2/x = y
(x^2+4)/2x=y
x^2-2xy+4=0
x^2-2xy+y^2-y^2+4=0
(x-y)^2-y^2+4=0
Since (x-y)^2 min = 0
=> y^2=4
=>y=2
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink] New post   31 Mar 2024, 13:57
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Re: If x is positive, what is the least possible value of x/2 + 2/x? [#permalink]
31 Mar 2024, 13:57
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