Question

Let f be a real valued function satisfying

f(x) + f(x + 6) = f(x + 3) + f(x + 9). Then $\int_{x}^{x + 12}{}$f(t) dt is

• A. A linear function of x
• B. An exponential function of x
• C. A constant function
• D. None of these

Answer 1

Answer: (C)

A constant function

Answer 2

f(x) + f(x + 6) = f(x + 3) + f(x+9) …(1)

x = x + 3

f(x + 3) + f(x + 9) = f(x + 6) + f(x + 12)

from eqⁿ (1)

f(x) + f(x + 6) – f(x + 9) + f(x + 9)

= f(x + 6) + f(x + 12)

f(x) = f(x + 12)

Let g(x) = $\int_{x}^{x + 12}{}$f(t) dt

g'(x) = $\left\lbrack f(t) \right\rbrack_{x}^{x + 12}$

g ' (x) = f(x + 12) – f(x) = 0

So g(x) is a constant function