The second derivative:

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{f"(x)}& =0\hspace{0.5em}\mathrm{for}\hspace{0.5em}x\le 0\\ \multicolumn{1}{c}{}& =4\hspace{0.5em}\mathrm{for}\hspace{0.5em}x>0.\end{array}}$

Hence the second derivative does not exist at the origin because on the left the limiting value of $f"(x)$ as $x\to 0$ is 0 whereas on the right the limiting value of $f"(x)$ as $x\to 0$ is 4.