Question

Let the functions $f:R\to R$ and $g:R\to R$ be defined as:
$f\left(x\right)=\{\begin{array}{cc}x+2,& x<0\\ x^2,& x\ge 0\end{array}\text{and}g\left(x\right)=\{\begin{array}{cc}{x}^3,& x<1\\ 3x-2,& x\ge 1\end{array}$
Then, the number of points in $R$ where $\left(fog\right)\left(x\right)$ is non differentiable is equal to :

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is A $1$

$fog\left(x\right)=\{\begin{array}{cc}{x}^3+2,& x<0\\ x^6,& 0\le x<1\\ (3x-2{)}^2,& x\ge 1\end{array}$
Clearly, $fog\left(x\right)$ is discontinuous at $x=0$ then non - differentiable at $x=0$
Now, at $x=1$
R.H.D.
$=\underset{h\to 0^{+}}{lim}\frac{f(1+h)-f\left(1\right)}{h}=\underset{h\to 0^{+}}{lim}\frac{\left(3\right(1+h)-2{)}^2-1}{h}=6$
L.H.D.
$=\underset{h\to 0^{-}}{lim}\frac{f(1-h)-f\left(1\right)}{-h}=\underset{h\to 0^{-}}{lim}\frac{(1-h{)}^6-1}{-h}=6$
Number of point of non - differentiability is $1.$