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?

Question

Question

Answer 1

Concept:

\({\rm{f}}\left( {\rm{x}} \right) = {{\rm{a}}^{\rm{x}}},{\rm{Then\;f'}}\left( {\rm{x}} \right) = {{\rm{a}}^{\rm{x}}}\ln {\rm{a}}\)

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) = 2^x

⇒ f’(x) = 2^xln 2

∴ f’(2) = 2² ln 2 = 4 ln 2

Hence, option (2) is correct.