Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.
Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.

• A
  Statement-$I$ is true,Statement-$II$ is true; Statement-$II$ is a correct explanation for Statement-$I$.
• B
  Statement-$I$ is true,Statement-$II$ is true; Statement-$II$ is not a correct explanation for Statement-$I$.
• C
  Statement-$I$ is true,Statement-$II$ is false.
• D
  Statement-$I$ is false,Statement-$II$ is true.