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Math Help - increasing and decreasing functions

  1. #1
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    increasing and decreasing functions

    ok here is the problem, consider a function f whose derivative is given by f'(x) = (x-4)^2 * e^(-x/2).

    a.) the function f is increasing on the interval?

    b.) the function f is concave up on the interval(s)

    c.) the function f is concave down on the interval(s)

    ok so what i dont get is how to find the critical points. i only see one in the problem and thats 4.? right? and to find the concave up or down i would apply the second derivative test? this problem gets really messy once i try to do the 2nd derivative test. there has to be an easier way. can anyone help me out?
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  2. #2
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    Quote Originally Posted by slapmaxwell1 View Post
    ok here is the problem, consider a function f whose derivative is given by f'(x) = (x-4)^2 * e^(-x/2).

    a.) the function f is increasing on the interval?

    b.) the function f is concave up on the interval(s)

    c.) the function f is concave down on the interval(s)

    ok so what i dont get is how to find the critical points. i only see one in the problem and thats 4.? right?

    yes, for f'(x)


    and to find the concave up or down i would apply the second derivative test? this problem gets really messy once i try to do the 2nd derivative test. there has to be an easier way. can anyone help me out?

    show your work for f''(x)
    ...
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  3. #3
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    Quote Originally Posted by skeeter View Post
    ...
    well for the 2nd derivative test, i would use the quotient rule. f"(x) = [(x-4)^2]/e^(x/2) = ok ok ok i got the critical points using the second derivative test. x = 4, 8 well the critical points for the function are 4 and 8. now i have to test them to see where the concave up and down is.
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  4. #4
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    ok wait lets go back to the original problem, the answer in the back of the book says the answer is (-infinity, infinity) this is where the function is increasing?
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  5. #5
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    Quote Originally Posted by slapmaxwell1 View Post
    ok wait lets go back to the original problem, the answer in the back of the book says the answer is (-infinity, infinity) this is where the function is increasing?
    yes ... (x-4)^2 \cdot e^{-x/2} \ge 0 for all x
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