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Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$, defined as

$$ f(x)=2 x^{3}-3 x^{2}-12 x+1$$

Which of the following statements is/are correct?

(Here, $\mathbb{R}$ is the set of real numbers.)

  1. $f$ has no global maximizer
  2. $f$ has no global minimizer
  3. $x=-1$ is a local minimizer of $f$
  4. $x=2$ is a local maximizer of $f$

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