Exercise 5.12

Question

Exercise 12: If $f(x)=|x|^3$, compute $f'(x)$, $f''(x)$ for all real $x$, and show that $f^{(3)}(0)$ does not exist.

analambanomenos · Accepted Answer

For $x>0$, $f(x)=x^3$, $f'(x)=3x^2$, $f''(x)=6x$, $f^{(3)}(x)=6$, and for $x<0$, $f(x)=-x^3$, $f'(x)=-3x^2$, $f''(x)=-6x$, $f^{(3)}(x)=-6$.
    \begin{align*}
        f'(0) &= \lim_{tightarrow0}\frac{f(t)}{t}=\lim_{tightarrow0}\bigl(\mathop{m sign}(t)t^2\bigr)=0 \
        f''(0) &= \lim_{tightarrow0}\frac{f'(t)}{t}=\lim_{tightarrow0}\bigl(\mathop{m sign}(t)3t\bigr)=0
    \end{align*}
    $f^{(3)}(0)$ does not exist for otherwise $f^{(3)}(x)$ would have a simple discontinuity at $x=0$, which cannot happen by the Corollary to Theorem 5.12.

Exercise 5.12

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Exercise 5.12

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