If `f(x+y)=f(x)f(y)` for all real x and y, `f(6)=3` and `f'(0)=10`, then f'(6) is

Options

A. 30
B. 13
C. 10
D. 0

Answer

A

Explanation

Since, `f'(0)=underset(xto0)lim(f(x)-f(0))/(x-0)=10`
`impliesunderset(hto0)lim(f(0+h)-f(0))/(h)=10`
`impliesf(0)(underset(hto0)lim(f(h)-1)/(h))=10" "...(i)`
`[becausef(0+h)=f(0)f(h),"given"]`
Now, `f(0)=f(0)f(0)`
`implies" "f(0)=1`
`:.` From Eq. (i)
`underset(hto0)lim(f(h)-1)/(h)=10" "...(ii)`
Now, `f'(6)=underset(hto0)lim(f(6+h)-f(6))/(h)`
`=underset(hto0)lim((f(h)-1)/(h))f(6)["from Eq. (ii)"]`
`=10xx3=30`