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Question

Let f(x+y) = f(x).f(y) for all x and y. Given that f(3) = 3 and f'(0)= 11. Then the value of f'(3) is ___ .


Solution

f(x+y)=f(x).f(y)f(x)=limh0f(x+h)f(x)h=limh0f(x).f(h)f(x)h=f(x).limh0f(h)1h......(i)
Since f(x) is finite at least one point,
limh0f(h)1h is finite
limh0f(h)1=0 limh0f(h)=1
Since f(x) is differentiable at x=0, it is also continuous 
limh0f(h)=f(0)=1f(0)=limh0f(0+h)f(0)h=limh0f(h)1hf(x)=f(x) limh0f(h)1h.....from (i)f(x)=f(x).f(0)f(3)=f(3).f(0)=11 f(3)=11×3=33    [f(0)=11]
 

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