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Homework Help: Maximum value(Tricky)

  1. Jul 13, 2008 #1
    1. The problem statement, all variables and given/known data
    If x belongs set of real numbers, find the maximum value of [tex]2(a-x)(x+\sqrt{x^{2}+b^{2}})[/tex]

    3. The attempt at a solution
    All that I could do was to try and diffrentiate the above expression but it yields a polynomial of degree 4. A hint to this question is that it belongs to the Quadratic Equations Chapter. And the answer to th above question is [tex]a^{2}+b^{2}[/tex]
    Please help me. I cant make head or tail out of it.
     
  2. jcsd
  3. Jul 13, 2008 #2

    tiny-tim

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    complete the square …

    Hi ritwik06! :smile:

    You have to complete the square (twice).

    Hint: start with the easy bit … the 2ax. :smile:
     
  4. Jul 13, 2008 #3
    Re: complete the square …

    I didnt get u tim. Coul u please be more explicit...
    thanks
     
  5. Jul 13, 2008 #4

    tiny-tim

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    Re: complete the square …

    Do you know what "complete the square" means?

    If you do … then try it! :smile:

    (If you don't, I'll show you an example.)
     
  6. Jul 13, 2008 #5
    Re: complete the square …

    I tried this:
    help me!!!
    [tex]b^{2}+2ax+2a(\sqrt{b^{2}+x^{2}})-(x+(\sqrt{b^{2}+x^{2}))^{2}[/tex]
    Hey, please help. Dont think that I havnt tried it at all. I have done all that I could. Thanks for the help.
     
  7. Jul 13, 2008 #6

    tiny-tim

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    Re: complete the square …

    ok … two hints:

    i] You have two squares to complete, and you need x2 in both of them … so you'll have to split up the -2x2, won't you? :wink:

    ii] You know what the answer is, so that gives you a pretty good clue as to what might be left over! :smile:

    (And, as I said, get rid of the 2ax first.)
     
  8. Jul 14, 2008 #7
    Re: complete the square …

    I have worked more on this problem but yet no results come my way;
    [tex]((a+\sqrt(x^{2}+b^{2}))^{2})-((x-a)^{2})-((x+\sqrt(x^{2}+b^{2}))^{2})[/tex]
     
  9. Jul 14, 2008 #8

    tiny-tim

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    Re: complete the square …

    ok …

    try [tex]2(a-x)(x+\sqrt{x^{2}+b^{2}}) = a^2\ -\ (a-x)^2\ -\ x^2\ +\ 2(a-x)\sqrt{x^{2}+b^{2}}[/tex] :smile:
     
  10. Jul 21, 2008 #9

    Borek

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    It is not a correct answer.

    But I must admit I have no idea how to complete these squares. Not that I am surprised, I am mathematically challenged. I have solved it by brute force, which was much easier then expected.
     
  11. Jul 21, 2008 #10

    tiny-tim

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    completing the square

    Hi Borek! :smile:

    Try adding [itex]b^2\ -\ b^2[/itex] to my last post. :smile:
     
  12. Jul 22, 2008 #11

    Borek

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    Arrgh. There is nothing like solving different question that was asked. I was looking for x such that the function given gets maximum value. That's for

    [tex]x = \frac {a^2-b^2} {2a}[/tex]

    but

    [tex]{a^2+b^2}[/tex]

    is a correct max value.

    Thanks TT :smile:
     
  13. Jul 24, 2008 #12
    Thanks a lot Tim.
    I had got my answer the day you had posted your second last reply.
    Thank you very very much. I am sorry for xpressing my gratitude a bit late. Thanks again.
     
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