(a) Let f(x) =x2, x 0x2 + 1, x> 0Show thatlimx0 f(x) = limx0+ f(x)but that f(0) does not

Chapter 4, Problem 42

(choose chapter or problem)

(a) Let

Show that

$\lim_{x\rightarrow0^-}f^{\prime}(x)=\lim_{x\rightarrow0^+}f^{\prime}(x)$

but that ${f}^{'}(0)$ does not exist.

(b) Let

$f(x)=\left\{\begin{array}{ll} x^{2}, & x \leq 2 \\ x^{3}, & x>0 \end{array}\right.$

Show that $f^{\prime}(0)$ exists but $f^{\prime \prime}(0)$ does not.

Equation Transcription:

$\lim\limits_{{x} \rightarrow {{0}^{-}}}f'(x)=\lim\limits_{{x} \rightarrow {{0}^{+}}}f'(x)$

$f(x)=$ { ${\ }_{{x}^{3},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ >\ 0}^{{x}^{2},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ \leq{}\ 2}$

${f}^{'}(0)$

${f}^{''}(0)$

Text Transcription:

Lim_x rightarrow 0^- f^prime(x) = lim_x rightarrow 0^+ f^prime(x)

f(x)={x^2, x leq 2 x^3, x>0

f^prime(0)

f^prime prime(0)

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