Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

• A
  Statements $(i)$ and $(ii)$ are false and $(iii)$ is true.
• B
  Statements $(i)$ and $(ii)$ are true and $(iii)$ is false.
• C
  Statements $(ii)$ and $(iii)$ are true and $(i)$ is false.
• D
  Statements $(i)$ and $(iii)$ are true and $(ii)$ is false.