Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :
$f(x)=\left\{\begin{array}{ll}x+2, & x<0 \\ x^{2}, & x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{lr}x^{3}, & x<1 \\ 3 x-2, & x \geq 1\end{array}\right.$
Then, the number of points in $\mathbb{R}$ where $(f \circ g)(x)$ is NOT differentiable is equal to :
Correct Option: , 2
$f(g(x))= \begin{cases}g(x)+2, & g(x)<0 \\ (g(x))^{2}, & g(x) \geq 0\end{cases}$
$= \begin{cases}x^{3}+2, & x<0 \\ x^{6}, & x \in[0,1) \\ (3 x-2)^{2}, & x \in[1, \infty)\end{cases}$
$(f \circ g(x))^{\prime}= \begin{cases}3 x^{2}, & x<0 \\ 6 x^{5}, & x \in(0,1) \\ 2(3 x-2) \times 3, & x \in(1, \infty)\end{cases}$
At 'O'
L.H.L. $\neq$ R.H.L. (Discontinuous)
At '1'
L.H.D. $=6=$ R.H.D.
$\Rightarrow f \circ g(x)$ is differentiable for $x \in \mathbb{R}-\{0\}$
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