question_answer

Directions: (1 - 5)

Let \[f\text{ }\left( x \right)\]be a real valued function, then its

Left Hand Derivative (L.H.D.) :

\[Lf'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a-h \right)-f\left( a \right)}{-h}\]

Right Hand Derivative (R.H.D.) :

\[Rf'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}\]

Also, a function \[f\left( x \right)\]is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal.

For the function  answer the following questions

R.H.D. of \[f\left( x \right)\] at \[x=1\] is

A) 1

B) -1

C) 0

D) 2

Correct Answer: B

Solution :

We have, \[Rf'\left( 1 \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( 1+h \right)-f\left( 1 \right)}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\,\frac{3-\left( 1+h \right)-2}{h}=\underset{h\to 0}{\mathop{\lim }}\,-\frac{h}{h}=-1\]