exercise:23efe261f4

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There are many examples of a function [math]f[/math] and a number [math]a[/math] such that [math]f(a)[/math] is defined ([math]a[/math] is in the domain of [math]f[/math]) but [math]f^\prime(a)[/math] does not exist. Another way of saying the same thing is that the domain of [math]f^\prime[/math] can be a ph proper subset of the domain of [math]f[/math]. It is equally possible for [math]f^\prime (a)[/math] to be defined and [math]f^{\prime\prime} (a)[/math] not to be. Let [math]f[/math] be the function defined by

[[math]] f(x) = \begin{cases}\frac{x^2}2 \mbox{if $x\geq 0$,} \\ -\frac{x^2}2 \mbox{if $x\leq 0$.}\end{cases} [[/math]]

• Compute [math]f^\prime[/math].
• Is [math]f[/math] a differentiable function?
• Show that [math]f^{\prime\prime}(0)[/math] does not exist, and compute [math]f^{\prime\prime}(x)[/math] for [math]x\ne 0[/math].