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If f(x+y)=2f(x) f(y) for all x,y where f...

If `f(x+y)=2f(x) f(y)` for all x,y where f'(0)=3 and f(4)=2, then f'(4) is equal to

A

6

B

12

C

4

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
We have
`f'(4)=underset(h to 0)lim (f(4+h)-f(4))/(h)`
`f'(4)=underset(h to 0)lim (f(4+h)-f(4+0))/(h)`
`f'(4)=underset(h to 0)lim ((2f(4)f(h)-2f(4)f(0))/(h))" "[{:(,f(x+y)),(,=2f(x)f(y)):}]`
`f'(4)=underset(h to 0)lim 2f(4) ((f(h)-f(0))/(h-0))`
`f'(4)=underset(h to 0)lim (f(h)-f(0))/(h-0)=4f'(0)=4xx3=12`
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