For the complex function
$$f(z)=f(x+i y)=x^{3}-4 x y^{2}+i\left(4 x^{2} y-2 y^{3}\right)$$
consider the following two statements
$P: f$ is not analytic at $(0,0)$.
$Q: f$ does not satisfy the Cauchy-Riemann equations along the $x$-axis.
Then, which one of the following statements is true?
- $P$ is false and $Q$ is true
- $P$ is true and $Q$ is false
- Both $P$ and $Q$ are true, and $Q$ implies $P$
- Both $P$ and $Q$ are true, but $Q$ does not imply $P$
