Functions Question 475
Question: Let $ f(x+y)=f(x)+f(y) $ and $ f(x)=x^{2}g(x) $ for all $ x,y\in R $ , where $ g(x) $ is continuous function. Then $ f’(x) $ is equal to
Options:
A) $ g’(x) $
B) $ g(0) $
C) $ g(0)+g’(x) $
D) 0
Show Answer
Answer:
Correct Answer: D
Solution:
We have $ f’(x)=\underset{h\to 0}{\mathop{\lim }},\frac{f(x+h)-f(x)}{h} $ $ =\underset{h\to 0}{\mathop{\lim }},\frac{f(x)+f(h)-f(x)}{h} $ $ [ \because f(x+y)=f(x)+f(y) ] $ $ =\underset{h\to 0}{\mathop{\lim }},\frac{f(h)}{h}=\underset{h\to 0}{\mathop{\lim }},\frac{h^{2}g(h)}{h}=0.g(0)=0 $ $ [\because g $ is continuous therefore $ \underset{h\to 0}{\mathop{\lim }},g(h)=g(0)] $ .
BETA
