Question

Let f be a real valued function satisfying f(x) + f(x + 4) = f(x + 2) + f(x + 6) then graph of the function
$g\left(x\right)=\stackrel{x+8}{\underset{x}{\int }}f\left(t\right)dt$ is

• A. an inclined line
• B. a curve which is strictly increasing
• C. a curve which is strictly decreasing
• D. a horizontal line

Answer 1

The correct option is D a horizontal line

Replace $x$ with $x+2$,then $f(x+2)+f(x+6)$
$=f(x+4)+f(x+8)=f\left(x\right)+f(x+4)$
$\Rightarrow f\left(x\right)=f(x+8)$
Hence $g^{\prime }\left(x\right)=f(x+8)-f\left(x\right)=0\Rightarrow g\left(x\right)$ is constant