Q: Let f be a function satisfying f(x + y) = f(x) + f(y) and f(x) = x2 g(x) for all x and y, where g(x) is a continuous function, then f ‘(x) is equal to
(A) g'(x)
(B) g(0)
(C) g(0) + g'(x)
(D) 0
Solution : $\large f'(x) = lim_{h \rightarrow 0 } \frac{f(x+h) – f(x)}{h}$
$\large = lim_{h \rightarrow 0 } \frac{f(x) + f(h) – f(x)}{h}$
$\large = lim_{h \rightarrow 0 } \frac{f(h)}{h} $
$\large = lim_{h \rightarrow 0 } \frac{h^2 g(x)}{h} = 0 $
Hence (D) is the correct answer.