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Q.

Let f  be a function satisfying f(x+y)=f(x)+f(y)  and f(x)=x2g(x)  for all x and y, where g(x) is a continuous function, then f'(x)  is equal to

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a

g'(x)

b

g(0)

c

g(0)+g'(x)

d

0

answer is D.

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Detailed Solution

f'(x)=limh→0f(x+h)−f(x)h =limh→0f(x)+f(h)−f(x)h                         (∵f(x+y)=f(x)+f(y))=limh→0f(h)h=limh→0h2g(h)h=0                (∵f(x)=x2g(x))
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