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Homework Help: Second derivative of sqrt(x) * e^(-x)

  1. Mar 25, 2009 #1
    1. The problem statement, all variables and given/known data
    find f''(x) if f(x) = sqrt(x) * e^(-x) and then find the roots of f''(x)

    // im trying to do the 2nd derivative test (need f''x) and then find inflection points//

    2. Relevant equations

    my methodology| d/dx sqrt(x) = 1/(2*sqrt(x)) and d/dx e^(-x) = -e^(-x)

    3. The attempt at a solution

    i found f'(x) to be: e^(-x) /(2*sqrt(x)) + (-e^(-x) *sqrt(x))

    and then f''(x) should be d/dx [ e^(-x) /(2*sqrt(x))] + d/dx [-e^(-x) *sqrt(x)]

    and ive gone through it a few times but what i get is:

    -e^(-x)*(2*sqrt(x)) - [2/(2*sqrt(x))] + [e^(-x)*sqrt(x) + -e^(-x)/(2*sqrt(x))]

    when i try to set this to zero i just get the feeling that my f'' is just completely wrong. When i first saw this problem i thought "no problem" but now i dont know. Is this problem covertly difficult, or am i doint something wrong? THANKS!
  2. jcsd
  3. Mar 25, 2009 #2
    Looks like your f'(x) is right.

    However, yeah, I'd say your f''(x) has some issues.

    Remember that 1/sqrt(x) = x^(-1/2). If you remember that, it should be easier to take your derivative, just remember to do the product rule again.

    This means that d/dx (1/sqrt(x)) = d/dx (x^(-1/2)) = -(1/2)x^(-3/2).

    Hope that helps a bit!
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