A function f: R R satisfies the relation f(x+y)=f(x) f(y), x, y R and f(x) 0 , x R . If f is differentiable at x=0 and f^ (0)=4 and f(6)=3 , then f^ (6) is equal to

Step-by-step solution

aligned & f: R R & f(x+y)=f(x) f(y), x, y R & f(x) 0, x R & and f^ (0)=4 and f(6)=3 & f(x+y)=f(x) f(y) ...(i) & Put x=0 y=0 & f(0)=f(0) f(0) & f(0)[f(0)-1]=0 & f(0)=0 and f(0)-1=0 aligned not possible and f(0)=1 aligned & f^ (0)=4 & _ h 0 f(0+h)-f(0) h =4 & _ h 0 (f(h)-1) h =4 ...(ii) & aligned Now, aligned f^ (x) & = _ h 0 f(x+h)-f(x) h & = _ h 0 f(x) f(h)-f(x) h & = _ h 0 f(x) [ f(h)-1 h ] & =f(x) _ h 0 f(h)-1 h & =f(x) 4[by Eq. (ii)] f^ (x) & =4 f(x) f^ (6) & =4 f(6)=4 3=12 aligned