Let the functions
Question:

Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $\mathrm{g}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :

$f(x)=\left\{\begin{array}{ll}x+2, & x<0 \\ x^{2}, & x \geq 0\end{array}\right.$ and $g(x)= \begin{cases}x^{3}, & x<1 \\ 3 x-2, & x \geq 1\end{cases}$

Then, the number of points in $\mathbb{R}$ where $(f \circ g)(x)$ is NOT differentiable is equal to :

 

  1. (1) 3

  2. (2) 1

  3. (3) 0

  4. (4) 2


Correct Option: , 2

Solution:

$f(g(x))= \begin{cases}g(x)+2, & g(x)<0 \\ (g(x))^{2}, & g(x) \geq 0\end{cases}$

$= \begin{cases}x^{3}+2, & x<0 \\ x^{6}, & x \in[0,1) \\ (3 x-2)^{2}, & x \in[1, \infty)\end{cases}$

$(f \circ g(x))^{\prime}= \begin{cases}3 x^{2}, & x<0 \\ 6 x^{5}, & x \in(0,1) \\ 2(3 x-2) \times 3, & x \in(1, \infty)\end{cases}$

At ‘O’

L.H.L. $\neq$ R.H.L. (Discontinuous) At ‘1’

L.H.D. $=6=$ R.H.D. $\Rightarrow \quad f \circ g(\mathrm{x})$ is differentiable for $\mathrm{x} \in \mathbb{R}-\{0\}$

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