Question

Let $g(x)$ is an even function and $f(x)=x-\sin x$. If $x=x_i$, where $i=1,2,3,\dots \dots ,5$ are the 5 distinct solutions of the equation $f(x)=g^{\prime }(x)$ and $\sum _{i=1}^{5}f\left(x_i\right)=k\pi$, then k equals

• A. 0
• B. 3
• C. 5
• D. 6

Answer 1

Answer: (A)

$f(x)=x-\sin x$ is an odd function
Given, $f(x)=g^{\prime }(x)$ has five distinct solution of the form of $0,\pm \alpha ,\pm \beta$ (think!)
$\therefore \sum _{i=1}^{5}f\left(x_i\right)=0\Rightarrow k=0$