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Math Help - Define domain and intervals of rising/falling

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    Member courteous's Avatar
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    Question Define domain and intervals of rising/falling

    "Define domain and intervals of rising/falling of the following function f(x): f(x)=\int_{0}^{x}\log(t^3+1)dt."

    Domain of f(x) is where t>-1: (-1, \infty)

    What about function falling/rising (assuming that I got the domain right)? Isn't it so, that you just take derivative of f(x) and that "cancels" the integral, which would mean that f(x) is always rising for t\geq 0, because its derivative ( \log(t^3+1)) is always positive with t\geq 0?
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  2. #2
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    Quote Originally Posted by courteous View Post
    "Define domain and intervals of rising/falling of the following function f(x): f(x)=\int_{0}^{x}\log(t^3+1)dt."

    Domain of f(x) is where t>-1: (-1, \infty)

    What about function falling/rising (assuming that I got the domain right)? Isn't it so, that you just take derivative of f(x) and that "cancels" the integral, which would mean that f(x) is always rising for t\geq 0, because its derivative ( \log(t^3+1)) is always positive with t\geq 0?
    f(x) is 'rising' for values of x such that f'(x) > 0 and 'falling' for values of x such that f'(x) < 0.

    Note that f'(x) = \log (x^3 + 1). And note that \log (x^3 + 1) > 0 for x > 0 \, ....
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