Question

Consider the following statements :
1. $f^{\prime }(2+0)$ does not exist.
2. $f^{\prime }(2-0)$ does not exist.
Which of the above statements is/are correct ?

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

Answer 1

Answer: (C)

Both 1 and 2

$f^{\prime }\left(2^{+}\right)=\underset{x\to 2^{+}}{lim}{f}^{\prime }\left(x\right)=\underset{x\to 2^{+}}{lim}3-2x=-1$

$f^{\prime }\left(2^{-}\right)=\underset{x\to 2^{-}}{lim}{f}^{\prime }\left(x\right)=\underset{x\to 2^{-}}{lim}-x=-2$

Hence, $f^{\prime }\left(2\right)$ does not exist and hence, both statements are correct.