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Let f: R->R be a differentiable funct...

Let `f: R->R` be a differentiable function with `f(0)=1` and satisfying the equation `f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y)` for all `x ,\ y in R` . Then, the value of `(log)_e(f(4))` is _______

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The correct Answer is:
2
Given, `f(x+y)=f(x)f'(y)+f'(x)f(y), AAx,y in R and f(0)=1`
Put x = y = 0, we get `f(0) = f(0) f'(0) + f'(0) f(0)`
`rArr 1 = 2f'(0) rArr f'(0)=(1)/(2)`
Put x = x and y = 0, we get `f(x) = f(x) f'(0) + f'(x) f(0)`
`rArr" "f(x) = (1)/(2) f(x) + f'(x)`
`rArr" "f'(x)=(1)/(2)f(x) rArr (f'(x))/(f(x))=(1)/(2)`
On integrating, we get `log f(x) =(1)/(2)x+C`
`rArr" "f(x)=Ae^((1)/(2)x), "where e"^(c) = A`
If `f(0) = 1, "then" A = 1`
`"Hence"," "f(x)=e^((1)/(2)x)`
`rArr" " log_(e)f(x) = (1)/(2)x`
`rArr" "log_(e)f(4)=(1)/(2) xx 4 = 2`
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