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Thread: Define domain and intervals of rising/falling

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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): $\displaystyle f(x)=\int_{0}^{x}\log(t^3+1)dt$."

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

    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 $\displaystyle t\geq 0$, because its derivative ($\displaystyle \log(t^3+1)$) is always positive with $\displaystyle t\geq 0$?
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    Quote Originally Posted by courteous View Post
    "Define domain and intervals of rising/falling of the following function f(x): $\displaystyle f(x)=\int_{0}^{x}\log(t^3+1)dt$."

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

    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 $\displaystyle t\geq 0$, because its derivative ($\displaystyle \log(t^3+1)$) is always positive with $\displaystyle t\geq 0$?
    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 $\displaystyle f'(x) = \log (x^3 + 1)$. And note that $\displaystyle \log (x^3 + 1) > 0 $ for $\displaystyle x > 0 \, ....$
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