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Maximum

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Lê Anh Duy
01/09/2018 at 16:02
Votes Answers Follow

28) Given x2 + 5y2 - 4xy - x + 2y - 6 = 0

Find the max and min values of A = x - 2y

MaximumMinimummemorable equality
  • ...
    Alone 23/10/2018 at 13:01

    \(x^2+5y^2-4xy-x+2y-6=0\)

    \(\Leftrightarrow x^2-4xy+4y^2-\left(x-2y\right)-6=-y^2\)

    \(\Leftrightarrow\left(x-2y\right)^2-\left(x-2y\right)-6=-y^2\le0\)

    \(\Leftrightarrow\left(x-2y+2\right)\left(x-2y-3\right)\le0\)

    \(\Leftrightarrow-2\le x-2y\le3\)

    Lê Anh Duy selected this answer.

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Mr Puppy
14/07/2018 at 06:21
Votes Answers Follow

Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treatrous and selfish (especially the captain). The captain always proposes a distribution of the loot. All pirates vote on the proposal and if at least half of the crew (which includes himself) approve, the loot will be divided as proposal, as no pirates would be willing to take on the captain without superior force on their side. Otherwise he will face a mutiny: all pirates will turn against him and make him walk the plank. The pirates will start over again with the next senior pirate as captain. What's the maximum number of coins the captain can keep without risking his life?

Maximum

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FoLder
26/07/2017 at 11:30
Votes Answers Follow

Let a,b,c > 0.

Find MAX of : \(A=\dfrac{abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

Maximum
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    Searching4You 26/07/2017 at 11:53

    Use Cauchy's inequality for positive numbes a,b,c.

    \(a+b\ge2\sqrt{ab},b+c\ge2\sqrt{bc},c+a\ge2\sqrt{ca}\)

    \(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge8\sqrt{ab}\cdot\sqrt{bc}\cdot\sqrt{ca}=8abc\)

    \(\Rightarrow A=\dfrac{abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\le\dfrac{1}{8}\)

    \(MaxA=\dfrac{1}{8}\Leftrightarrow a=b=c>0\)

    Selected by MathYouLike
  • ...
    ¤« 03/04/2018 at 13:32

    Use Cauchy's inequality for positive numbes a,b,c.

    a+b≥2ab−−√,b+c≥2bc−−√,c+a≥2ca−−√

    ⇒(a+b)(b+c)(c+a)≥8ab−−√⋅bc−−√⋅ca−−√=8abc

    ⇒A=abc(a+b)(b+c)(c+a)≤18

    MaxA=18⇔a=b=c>0


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Summer Clouds moderators
22/07/2017 at 12:01
Votes Answers Follow

Given \(H=\dfrac{1}{x^2-2x+5}\). Find the value of x such that H reaches the maximum value.

Maximum
  • ...
    Kayasari Ryuunosuke Coodinator 22/07/2017 at 12:10

    We have :

    \(x^2-2x+5=x^2-2x+1+4\)

                          \(=\left(x-1\right)^2+4\ge4\)

    \(\Rightarrow H=\dfrac{1}{x^2-2x+5}\le\dfrac{1}{4}\)

    \(\Rightarrow Max_H=\dfrac{1}{4}\)

    \(\Leftrightarrow\left(x-1\right)^2=0\)

    \(\Leftrightarrow x=1\)

    Selected by MathYouLike
  • ...
    ¤« 03/04/2018 at 13:33

    We have :

    x2−2x+5=x2−2x+1+4

                          =(x−1)2+4≥4

    ⇒H=1x2−2x+5≤14

    ⇒MaxH=14

    ⇔(x−1)2=0

    ⇔x=1


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Dung Trần Thùy
19/03/2017 at 22:06
Votes Answers Follow

Find the maximum and the minimum of \(N=\dfrac{4x-8}{x^2-4x+8}\).

MaximumMinimumFraction
  • ...
    ¤« 03/04/2018 at 13:33

    (*) 1+N=.....=x2/(x-2)2+4

    We have x2 ≥

     0 , (x-2)2+4 ≥

     4 > 0 

    So 1+N ≥

     0 => N ≥

     -1 ;equality : x=0 

    (*)1-N=....=(x-4)2/(x-2)2+4 

    ....... -> N ≤

     1 , equality : x=4

  • ...
    ¤« 03/04/2018 at 13:32

    There is something is wrong here 

    Change to :

    We have a2+b2≥5

    => 9−(a2+b2)≤4

    => 2ab≤4

    => ab≤2

    <=> a2b2≥4

    => 4a2b2≥16

  • ...
    Phan Văn Hiếu 28/03/2017 at 21:30

    trình bày = tiếng việt đc ko tiếng anh ngại viết lắm


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