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-   -   Derivative of exponential functions (https://www.askmehelpdesk.com/showthread.php?t=236546)

  • Jul 12, 2008, 03:35 AM
    clueless123
    Derivative of exponential functions
    Hi there, I'm having difficulties with this particular question

    The function h: R- > R, h(t)=(t^2+at+10) e ^ - t, where a is a real constant, has a derivative h'(t) = (-t^2 + (2-a)t + (a-10))e^ -t.

    Find the values of a such that
    I) the graph of y=h(t) has exactly on stationery point.
    ii) h'(t) <0 for all t is an element of R.

    This is confusing as I don't know how to go about this question.
  • Jul 12, 2008, 07:44 AM
    galactus
    They give you the derivative in the form of a quadratic. The thing to do is use the

    discriminant, set it to 0 and solve for a. That will be the points where there is one root of

    multiplicity 2 and, therefore, one stationary point.

    The discriminant is

    We have from h'(t):

    So, we have:



    Solve for a.

    If the discriminant is <0, then it has no real roots.

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