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Math Help - inverse

  1. #1
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    inverse

    For what parameter m the function:
    f(x) = (m+2)x^3-3mx^2+9mx-1 is invertible function.
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  2. #2
    MHF Contributor red_dog's Avatar
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    f must be strictly increasing or decreasing.
    That means f'(x)>0 or f'(x)<0, \ \forall x\in\mathbf{R}

    f'(x)=3[(m+2)x^2-2mx+3m]\Rightarrow (m+2)x^2-2mx+9m\neq 0, \ \forall x\in\mathbf{R}

    \Delta =b^2-4ac=-8m(m+3)

    \Delta<0\Rightarrow m\in(-\infty,-3)\cup(0,\infty)

    In this case f is injective.

    If m\in(-\infty,-3)\Rightarrow\lim_{x\to -\infty}f(x)=\infty, \ \lim_{x\to\infty}f(x)=-\infty

    If m\in(0,\infty)\Rightarrow\lim_{x\to-\infty}f(x)=-\infty, \ \lim_{x\to\infty}f(x)=\infty

    In both cases f is continuous, so f(\mathbf{R})=\mathbf{R} and f is surjective.
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