Let $f:[-1,1] \rightarrow \mathbb{R}$ be a function such that $|f(x)| \leq x^2$ for all $x \in[-1,1]$. Then

• A

  $f^{\prime}(0)$ exists and $f^{\prime}(0) \lt 0$
• B

  $f^{\prime}(0)$ does not exist
• Correct answer
  C

  $f^{\prime}(0)$ exists and $f^{\prime}(0)=0$
• D

  $f^{\prime}(0)$ exists and $f^{\prime}(0) \gt 0$

Hence, the correct answer is option: 'C'.