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Please Help Using the Second Derivative Test

  1. Oct 22, 2007 #1
    1. The problem statement, all variables and given/known data
    Find all relative extrema using the second derivative test for H(x) = x * lnx


    2. Relevant equations



    3. The attempt at a solution
    H'(x) = (1 * ln x) + (x * 1/x) = lnx + 1
    H''(x) = 1/x + 0

    Is H''(x) right? Then I am having trouble finding the relative extrema from the second derivative test?
     
  2. jcsd
  3. Oct 22, 2007 #2
    h''(x)=1/x
     
  4. Oct 22, 2007 #3
    Okay so how do I solve for the relative extrema?
     
  5. Oct 22, 2007 #4
    just try graphing the function and go from there. i dont remember what the second derevative test is, its been a long time. dont you have to set denominator =0 and solve x?if so x=0.
     
  6. Oct 22, 2007 #5

    Dick

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    Set H'(x)=0 and solve for x to find the critical points. For each critical point x test the sign of H''(x) to see if it's a max or a min.
     
  7. Oct 23, 2007 #6
    H"(x) = 0
    I am having trouble finding the relative extrema from the second derivative test?
     
  8. Oct 23, 2007 #7

    Dick

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    Set H'(x)=0 not H''(x)=0. Critical points are where H'(x)=0.
     
  9. Oct 23, 2007 #8
    How do you solve lnx + 1 = 0 for x????
     
  10. Oct 23, 2007 #9
    would it be...
    lnx + 1 = 0
    lnx = -1
    x = e^-1
     
  11. Oct 23, 2007 #10
    Then I would plug that into H" = 1/x, H" = 1/(e^-1) = e > 0 so its a relative max???
     
  12. Oct 23, 2007 #11

    Dick

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    That's almost all exactly correct, except the final conclusion. How can you say H''(x)>0 at a critical point means it's a max?? Don't you have like a textbook or something?
     
  13. Oct 24, 2007 #12
    H''(e^-1) > 0 means a relative max...thats what our book says to write? What do you mean? Is that wrong?
     
  14. Oct 24, 2007 #13

    HallsofIvy

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    Either you should throw away your textbook or you should read it more carefully- that's exactly backwards!
     
  15. Oct 30, 2007 #14
    Yea, for the second derivative tests things are sort of backwards.
    If f"(c)<0, then x=c is a relative maximum.
    If f"(c)>0, then x=c is a relative minimum.
    f f"(c)=0 or undefined, then the second derivative test is inconclusive.
     
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