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Let f(x + y) = f(x) f(y) and f(2) = 4 for all x, y ϵ R, where f(x) is continuous function. What is f' (2) equal to?

  1. 2 ln 2
  2. 4 ln 2
  3. ln 8
  4. 2 ln 3

Answer (Detailed Solution Below)

Option 2 : 4 ln 2
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Detailed Solution

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Concept:

Calculation:

Given that:  

f(2) = 4 and f (x + y) = f(x) f(y)

Putting x = 1, y = 1

f(2) = f(1) f(1) = 4

f(1) = f(1) = 2

i.e., f(1) = 21

Putting, x = 1, and y = 2

f (3) = f(1) f(2) = 2 × 4 = 23

∴ f(x) = 2x

⇒ f’(x) = 2x ln 2

∴ f’(2) = 22 ln 2 = 4 ln 2

Hence, option (2) is correct.

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