Question

Let $g(x)$ is an even function and $f(x)=x-\sin x$. If $x=x_i$, where $i=1,2,3, \ldots \ldots, 5$ are the 5 distinct solutions of the equation $f(x)=g^{\prime}(x)$ and $\displaystyle\sum_{i=1}^5 f\left(x_i\right)=k \pi$, 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!)
$\therefore \displaystyle\sum_{ i =1}^5 f \left( x _{ i }\right)=0 \Rightarrow k =0$