Question

Let $f$ be a real valued function satisfying $f\left(x\right)+f(x+4)=f(x+6)+f(x+2)$ then ${\int }_x^{x+8}f\left(z\right)dz$ is a/an

• A. constant function
• B. an odd function
• C. an even function
• D. periodic function

Answer 1

Answer: (A), (C), (D)

The correct options are
A constant function
C an even function
D periodic function

$f\left(x\right)+f(x+4)=f(x+2)+f(x+6)$
Substitute $x=x+2$
$f(x+2)+f(x+6)=f(x+4)+f(x+8)\Rightarrow f\left(x\right)=f(x+8)$
Let $g\left(x\right)={\int }_x^{x+8}f\left(t\right)dt$
$\Rightarrow g^{\prime }\left(x\right)=f(x+8)-f\left(x\right)=0$
$\Rightarrow g\left(x\right)$ is a constant function