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An integral question

  1. Aug 25, 2007 #1


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    Gold Member

    1. The problem statement, all variables and given/known data
    [tex]f(x) = \int^{x+1}_{x} e^{-t^2} dt[/tex]
    1) Where does f have a derivative?
    2) Prove that f(x)>0 for all x in R.
    3) Find the segments of R where f goes up and down and the extream points.

    2. Relevant equations

    3. The attempt at a solution

    1) The integral of a continues function has a derivative everywhere.

    2) [tex]e^{-t^2} [/tex] has a miniumum (m) in [x,x+1] and so
    [tex]m(x+1 -x) = m <= \int^{x+1}_{x} e^{-t^2} dt[/tex]
    But e^x > 0 for all x and so m>0.

    3)If [tex]F(x) = \int^{x}_{0} e^{-t^2} dt[/tex]
    Then f(x) = -F(x) + F(x+1) and so
    [tex]f'(x) =-e^{-x^2} + e^{-(x+1)^2} = e^{-x^2} ( e^{-(2x+1)} -1)[/tex]
    And so f'(x)=0 only when x= -1/2. For all x>-1/2 the f'(x) < 0 and the function goes down, and for all x<-1/2 f'(x)>0 and the function goes up.

    Is that right?
  2. jcsd
  3. Aug 25, 2007 #2


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    Homework Helper

    I don't see any problems.
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