Exercise 5.16

Question

Exercise 16:
    Suppose $f$ is twice-differentiable on $(0,\infty)$, $f''$ is bounded on $(0,\infty)$, and $f(x)\rightarrow 0$ as $x\rightarrow\infty$. Prove that $f'(x)\rightarrow 0$ as
    $x\rightarrow\infty$.

analambanomenos · Accepted Answer

Following the hint, let $f''$ be bounded by $M$ on $(0,\infty)$. Then by Exercise 15, $$\sup_{x>a}\big|f'(x)\big|\le4M\sup_{x>a}\big|f(x)\big|$$ which tends to 0 as
    $a\rightarrow\infty$.

Exercise 5.16

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Exercise 5.16

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