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Two functions f: R -> R and g: R -> R satisfy the relations f(x + y) = f(x) + f(y) + f(x)*f(y) and f(x) = x*g(x) for all x, y in R. It is also given that lim(x->0) g(x) = 1. Which of the following statements are TRUE?

1. f is differentiable at every x in R
2. If g(0) = 1, then g is differentiable at every x in R
3. The derivative f'(1) is equal to 1
4. The derivative f'(0) is equal to 1

Correct answer: The derivative f'(0) is equal to 1

Solution

Setting x=y=0 in the functional equation gives f(0) = 0. The derivative f'(x) = (1 + f(x))*lim[h->0](f(h)/h) = (1+f(x))*g(0). Since g(0) need not equal 1 in general, but lim g(x) as x->0 = 1 means f'(0) = 1 (using f(0)=0). Both A and D are true; C gives f'(1) = (1+f(1)) which depends on f(1). The JEE answer identifies A and D as correct.

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