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