Limits Continuity And Differentiability Question 106
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)-(x)}{h} $
$ =\underset{h\to 0}{\mathop{\lim }}\frac{f(x)+f(h)-f(x)}{h} $
$ =\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
