Limits Continuity And Differentiability Question 107
Question: Given $ f:[-2a,2a]\to R $ is an odd function such that the left hand derivative at x = a is zero and $ f(x)=f(2a-x)\forall x\in (a,2a), $ then its left had derivative at $ x=-a $ is
Options:
A) 0
B) a
C) -a
D) Does not exist
Show Answer
Answer:
Correct Answer: A
Solution:
Given $ f’(a)=\underset{h\to 0}{\mathop{\lim }}\frac{f(a-h)-f(a)}{-h}=0….(1) $ Now $ f’(-{a^{-}})=\underset{h\to 0}{\mathop{\lim }}\frac{f(-a-h)-f(-a)}{-h} $
$ =\underset{h\to 0}{\mathop{\lim }}\frac{-f(a+h)+f(a)}{-h} $ [ $ \because f(x) $ is odd function] $ =\underset{h\to 0}{\mathop{\lim }}\frac{-f(a-h)+f(a)}{-h} $
$ [\because f(2a-x)=f(x)\Rightarrow f(a+x)=f(a-x)] $
$ =\underset{h\to 0}{\mathop{\lim }}\frac{f(a-h)-f(a)}{h}=0 $ [From (1)]
BETA
