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Inequality of arithmetic and geometric means

  1. Jan 12, 2013 #1
    Hey! I have this: 2(√(1-a^2 ))+ 2a
    How to determine the maximum value of this?

    I think good for this is Inequality of arithmetic and geometric means, but I don't know how use this, because I don't calculate with this yet.

    So, have you got any ideas?



    Poor Czech Numeriprimi... If you don't understand my primitive English, write, I will try to write this better ;-)
     
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  3. Jan 12, 2013 #2

    symbolipoint

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    Could be a Calculus derivative problem. Does this f(a)=2(√(1-a^2 ))+ 2a have a maximum or minimum?
     
  4. Jan 12, 2013 #3

    HallsofIvy

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    Carrying that farther [itex]f(a)= 2(1- a^2)^{1/2}+ 2a[/itex] has derivative [itex] f'(a)= -2(1- a^2)^{-1/2}+ 2[/itex]. Any maximum (or minimum) must occur where that derivative is 0 or does not exist. It does not exist when a= 1 or -1 because in that case we have a 0 in the denominator. It is 0 when [itex]\sqrt{1- a^2}= 1[/itex] or a= 0.

    Check the values of [itex]2\sqrt{1- a^2}+ 2a[/itex] at x= -1, 0, and 1 to see which is the maximum value.

    Another way to look at this is to see that if [itex]y= 2\sqrt{1- a^2}+ 2a[/itex] then [itex]y- 2a= 2\sqrt{1- a^2}[/itex] and so [itex]y^2- 4ay+ 4a^2= 4- 4a^2[/itex] or [itex]8a^2- 4ay+ y^2= 4[/itex] which is the graph of an ellipse with major and minor axes rotated from the coordinate axes.
     
  5. Jan 12, 2013 #4
    Please, derivate no... I still have a few years time to learn this.
    I need to use Inequality of arithmetic and geometric means, bud how?
     
  6. Jan 12, 2013 #5

    AlephZero

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    Your expression has two parts, ##2\sqrt{1-a^2}## amd ##2a##.

    If you are supposed to use arithmetic and geometric means, you need to find some quantities that give those arithmetic and geometric means.

    For example if the quantities are ##x## and ##y## and the arithmetic mean is ##2a##, you have the equation ##(x+y)/2 = 2a##

    And you have another equation for the geometric mean .....
     
  7. Jan 13, 2013 #6
    Yes, i have this: n=4
    And what now?
     
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