Functions Question 542
Question: If $ f(x)= \begin{cases} & x+2,-1 \lt x \lt 3 \\ & 5,x=3 \\ & 8-x,x>3 \\ \end{cases} $ , then at $ x=3 $ , $ f’(x)= $
[MP PET 2001]
Options:
A) 1
B) - 1
C) 0
D) Does not exist
Show Answer
Answer:
Correct Answer: D
Solution:
If $ f(x)= \begin{cases} & x+2,,,-1<x<3 \\ & 5,x=3 \\ & 8-x,,x>3 \\ \end{cases} . $ and $ f(3)=5 $ L.H.D = $ \underset{x\to 3-}{\mathop{\lim }},\frac{f(x)-f(3)}{x-3}=\underset{h\to 0}{\mathop{\lim }},\frac{f(3-h)-f(3)}{-h} $ $ =\underset{h\to 0}{\mathop{\lim }},\frac{(3-h+2)-5}{-h}=\underset{h\to 0}{\mathop{\lim }},\frac{-h}{-h}=1 $ R.H.D $ =\underset{x\to {3^{+}}}{\mathop{\lim }},\frac{f(x)-f(3)}{x-3}=\underset{h\to 0}{\mathop{\lim }},\frac{f(3+h)-f(3)}{h} $ $ =\underset{h\to 0}{\mathop{\lim }},\frac{8-(3+h)-5}{h}=\underset{h\to 0}{\mathop{\lim }},\frac{-h}{h}=-1 $ L.H.D $ \ne $ R.H.D $ f(x) $ is not differentiable.
BETA
