Answer
The point $(0,0)$ is an inflection point, but $g''(0)$ does not exist.
Work Step by Step
$g(x) = x~\vert x \vert$
$g(x) = x^2~~~~~$ for $x \geq 0$
$g(x) = -x^2~~~~~$ for $x \lt 0$
$g'(x) = 2x~~~~~$ for $x \geq 0$
$g'(x) = -2x~~~~~$ for $x \lt 0$
$g''(x) = 2~~~~~$ for $x \gt 0$
$g''(x) = -2~~~~~$ for $x \lt 0$
However, $g''(0)$ does not exist since $\lim\limits_{h \to 0^-} g''(0+h) \neq \lim\limits_{h \to 0^+} g''(0+h)$
The graph is concave downward when $x \lt 0$
The graph is concave upward when $x \gt 0$
The graph changes from concave downward to concave upward at the point $(0,0)$,
The point $(0,0)$ is an inflection point, but $g''(0)$ does not exist.
