Question

Consider the following functions :
1. $f\left(x\right)=\{\begin{array}{ccc}\frac{1}{x}& if& x\ne 0\\ 0& if& x=0\end{array}$
2. $f\left(x\right)=\{\begin{array}{ccc}2x+5& if& x>0\\ x^2+2x+5& if& x\le 0\end{array}$
Which of the above functions is/are derivable at $x=0$?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is B 2 only

Solution:

1. Consider, $f^{\prime }\left(0\right)$

$=\underset{h\to 0}{lim}\frac{f(0+h)-f\left(0\right)}{h}=\underset{h\to 0}{lim}\frac{1/h-0}{h}$

$=\underset{h\to 0}{lim}\frac{1}{h^2}=\infty$ (not defined)

Hence, 1 is not differentiable at $x=0$

2. We have, $f\left(x\right)=\{\begin{array}{ccc}2x+5& if& x>0\\ x^2+2x+5& if& x\le 0\end{array}$

$⟹$$f^{\prime }\left(x\right)=\{\begin{array}{ccc}2& if& x>0\\ 2x+2& if& x\le 0\end{array}$

$\because LHL$ of $f^{\prime }\left(x\right)=RHL$ of $f^{\prime }\left(x\right)$

$\therefore f$ is differential at $x=0.$

Hence, B is the correct option.