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Upper envelope of a function

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data
    find the upper envelope for the family of ballistic curves :

    y = ax - [x^2(a^2+1)]/2

    an upper envelope is a curve y=F(x) such that for each x fixed, F(x) is the maximum of g(a) = ax - [x^2(a^2+1)]/2 for a in R


    2. Relevant equations



    3. The attempt at a solution
    diff g(a) wrt to a and equate to 0 and get a=1/x so F(x) is 1/x?? :(
     
  2. jcsd
  3. Feb 24, 2010 #2
    You are right to start by finding the maximum of [tex]g_x(a) = ax - \frac12 x^2 (a^2 + 1)[/tex], but you need to prove that the maximum of [tex]g_x[/tex] occurs at the single critical point you find.

    However, you have found the value of [tex]a[/tex] which maximizes [tex]g_x(a) = F(x)[/tex] at that [tex]x[/tex] --- not what you want, which is the value of the function at that point, expressed as a function of [tex]x[/tex]. What further computation is necessary here?
     
  4. Feb 24, 2010 #3
    Am I right in thinking you then have to sub the a=1/x back into the original y to get,

    y=F(x)=(1/2) x^2

    and mathmathmad your a warwick student right?
     
  5. Feb 24, 2010 #4
    I do not think so as it is not related with F(x) being maximum of g(a). Btw blackscorpian, I've sent you a pm.
     
  6. Feb 25, 2010 #5
    is it F(x)=1/2 - x^2/2 instead?

    @blackscorpion : yes... I am. you too?
     
  7. Feb 25, 2010 #6
    Yes it is
     
  8. Feb 25, 2010 #7
    Ahhh crap, it is aswell.
    Stupidly cancelled the ones forgettin bout the over 2 part.
    Well thats a mark thrown away, lol
     
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