Question

Let $f\left(x\right)=x\sin x$ and $g\left(x\right)=f\left(x\right)f^{\prime }\left(x\right)$,then number of distinct real roots of equation $g\left(x\right)=0wherexϵ(-2\pi ,2\pi )$is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C $7$

Given $f\left(x\right)=x\sin x$

$\Rightarrow f^{{}^{\prime }}\left(x\right)=x\cos x+\sin x$

$\Rightarrow g\left(x\right)=x\sin x(x\cos x+\sin x)$

$\Rightarrow g\left(x\right)=0\Rightarrow x\sin x(x\cos x+\sin x)=0$

$\Rightarrow x\sin x=0$ or $\tan x=-x$

(1)$x\sin x=0\Rightarrow x=\{-\pi ,0,\pi \}$

(2)$\tan x=-x$

To solve above equation, draw $\tan x$ graph and $y=-x$ graph in the interval $(-2\pi ,2\pi )$ and find out number of intersections

We get 5 intersections for above (see diagram)

Here '0' is a repeated value in both cases

$\therefore$total solutions $=3+4=7$