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Let f(x y)=f(x) f(y) for all x, y R . If f^ (1)=2 and f(4)=4 , then f^ (4) equal to

Options

  1. A4
  2. B1
  3. C1 2
  4. D8

Correct answer

D. 8

Step-by-step solution

We have f(x y)=f(x) f(y) for all x, y R . Putting x=y=1 , we get f(1)=f(1) f(1) f(1)[1-f(1)]=0 f(1)=1 [ f(1) 0] (i) Now, f^ (1)=2 _ h 0 f(1+h)-f(1) h =2 f(1) _ h 0 f(h)-1 h =2 _ h 0 f(1) f(h)-f(1) h =2 [ using f(1)=1] ...(ii) _ h 0 f(h)-1 h =2 Now f^ (4)= _ h 0 f(4+h)-f(4) h = _ h 0 f(4) f(h)-f(4) h = _ h 0 f(h)-1 h f(4)=2 f(4) from (i) =2 4=8

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