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  • Let f(x) be a function such that f(x+y) = f(x) + f(y) and f(x) = sin x g(x) for all x, y R. If g(x)&nbsp

Let f(x) be a function such that f(x+y) = f(x) + f(y) and f(x) = sin x g(x) for all x, y \in R. If g(x) is a continuous function such that g(0) = K, then,f "(x) is equal to.

Option: 1

K


Option: 2

Kx


Option: 3

K g(x)


Option: 4

None of these.


Answers (1)

best_answer

\mathrm{f^{\prime}(x) =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} }
           \mathrm{ =\lim _{h \rightarrow 0} \frac{f(x)+f(h)-f(x)}{h} }         \mathrm{[\because f(x+y)=f(x)+f(y)] }
           \mathrm{ =\lim _{h \rightarrow 0} \frac{f(h)}{h}=\lim _{h \rightarrow 0} \frac{\sin h g(h)}{h} }
           \mathrm{ =\lim _{h \rightarrow 0} \frac{\sin h}{h} \lim _{h \rightarrow 0} g(h)=g(0)=K} .

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Nehul

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