Question

Let $\frac{f(x + y)}{2}$= $\frac{f(x) + f(y)}{2}$, f '(0) = a and f(0) = b then f " (x) =

• A. 0
• B. a
• C. b
• D. a+b

Answer 1

Answer: (A)

0

Answer 2

Assume x → constant

y → variable

diff. equation

$\frac{1}{2}f'\left( \frac{x + y}{2} \right) = \frac{1}{2}f'(y)$

but y = 0 ; f '(x/2) = a

Integrate

f(x/2) = a(x/2) + C ⇒ f(x) = ax + c

put x = 0, f(0) = b ⇒ c = b

∴ f(x) = ax = b

f"(x) = 0 \end{matrix}$$