Sign of a differentiable function

Let $f : (-1, 1) \to \mathbb{R}$ be a differentiable function. Consider the following statements: (I) Suppose $f(0) \ge 0$, and $f'(x) > 0$ whenever $f(x) = 0$, for any $x \ge 0$. Then $f(x) > 0$, for any $x > 0$. (II) Suppose $f(0) \le 0$, and $f'(x) > 0$ whenever $f(x) = 0$, for any $x \le 0$. Then $f(x) < 0$, for any $x < 0$. Which of the following statements is correct?