Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by
$$
f(x)=\left\{\begin{array}{rc}
x^5+5 x^4+10 x^3+10 x^2+3 x+1, & x<0 \\
x^2-x+1, & 0 \leq x<1 \\
\frac{2}{3} x^3-4 x^2+7 x-\frac{8}{3}, & 1 \leq x<3 \\
(x-2) \log _c(x-2)-x+\frac{10}{3}, & x \geq 3
\end{array}\right.
$$
Then which of the following options is/are correct?
Select ALL correct options:
A
$f^{\prime}$ has a local maximum at $\mathrm{x}=1$
B
$f$ is onto
C
$f$ is increasing on $(-\infty, 0)$
D
$f^{\prime}$ is NOT differentiable at $\mathrm{x}=1$
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