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Let f x be a differentiable function on x ∈ R such that f x + y = f x ⋅ f y for all x , y . If f 0 ≠ 0 , f 5 = 12 and f ′ 0 = 16 , then f ′ 5 is equal to

Options

  1. A190
  2. B186
  3. C196
  4. D192

Correct answer

D. 192

Step-by-step solution

f ′ 5 = lim h → 0 f 5 + h − f 5 h = l i m h → 0 f 5 + h - f 5 + 0 h = l i m h → 0 f 5 · f h - f 5 · f 0 h ∵ f x + y = f x . f y f o r a l l x , y = l i m h → 0 f h - f 0 h · f 5 f ′ 0 × f 5 = 16 × 12 = 192

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