f(x+y)=f(x).f(y)andf(5)=2 f'(0)=3 then f'(5)=? for all value of'x'and'y'

Dear Praveen

Put x=y=0 the f(0)=f2(0)

f(0)=0 or 1

f'(0)=lim h->0 (f(0+h)-f(0))/h= (f(0)f(h)-f(0))/h= f(0)*(f(h)-1)/h

now f(0) cant be zero as f'(0) is 3

f(0)=1

lim h->0 (f(h)-1)/h = f'(0)=3  .............(1)

f'(5)=lim h->0 (f(5+h)-f(5))/h= (f(5)f(h)-f(5))/h= f(5)*(f(h)-1)/h

f'(5)=f(5)*3=2*3=6     using (1)

All the best.                                                           

AKASH GOYAL

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Approved

Dear Student,
Please find below the solution to your problem.

Put x=y=0 the f(0)=f2(0)
f(0)=0 or 1
f'(0)=lim h->0 (f(0+h)-f(0))/h= (f(0)f(h)-f(0))/h= f(0)*(f(h)-1)/h
now f(0) cant be zero as f'(0) is 3
f(0)=1
lim h->0 (f(h)-1)/h = f'(0)=3 .............(1)
f'(5)=lim h->0 (f(5+h)-f(5))/h= (f(5)f(h)-f(5))/h= f(5)*(f(h)-1)/h
f'(5)=f(5)*3=2*3=6 using (1)

Thanks and Regards