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Math Help - Increasing and decreasing interval, natural exponential function

  1. #1
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    Increasing and decreasing interval, natural exponential function

    Given: f(x) = e^(-2x^2)

    f'(x) evaluated as -4xe^(-2x^2)

    Now, every example I've seen thus far is a polynomial function, which makes it easy to find the roots and determine what interval it's increasing or decreasing, not so here. Any tips?
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    Quote Originally Posted by angrynapkin View Post
    Given: f(x) = e^(-2x^2)

    f'(x) evaluated as -4xe^(-2x^2)

    Now, every example I've seen thus far is a polynomial function, which makes it easy to find the roots and determine what interval it's increasing or decreasing, not so here. Any tips?
    The solution to -4x e^{-2x^2} = 0 is x = 0 since e^{-2x^2} \neq 0 for real values of x.

    Note also that e^{-2x^2} > 0 for all real values of x. So f'(x) > 0 => -4x > 0 and f'(x) < 0 => -4x < 0 ....
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    mr fantastic, thank you for your reply. I've worked through the problem and studied notes a little more carefully. The second derivative test returns concave upward for all real values, which is correct according to the graph of the original function.

    f"(x) = 4e^(2x^2)[(1+4x^2)]

    I set 1+4x^2 to zero as my two critical points and tested away.

    Not too sure how you evaluated 4xe^(2x^2) = 0

    From the graph, I know it has to be zero, but algebraically I'm not quite sure how to work that out (other than the assumption it HAS to equal zero)
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    Quote Originally Posted by angrynapkin View Post
    mr fantastic, thank you for your reply. I've worked through the problem and studied notes a little more carefully. The second derivative test returns concave upward for all real values, which is correct according to the graph of the original function.

    f"(x) = 4e^(2x^2)[(1+4x^2)]

    I set 1+4x^2 to zero as my two critical points and tested away.

    Not too sure how you evaluated 4xe^(2x^2) = 0

    From the graph, I know it has to be zero, but algebraically I'm not quite sure how to work that out (other than the assumption it HAS to equal zero)
    If A B = 0 then either A = 0 or B = 0 .....
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    Ok thanks, if something similar pops up on the final I just wanted something like that to write out.
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