Question

If the equation ${\cos }^{-1}\left(\cos x\right)=\frac{n-x}{n}$, has seven roots for all $x\ge 0$, then $n\in$

• A. $(0,2\pi )$
• B. $(2\pi ,4\pi )$
• C. $(4\pi ,6\pi )$
• D. $(6\pi ,8\pi )$

Answer 1

The correct option is D $(6\pi ,8\pi )$

${\cos }^{-1}\left(\cos x\right)=\frac{n-x}{n}$
$\Rightarrow {\cos }^{-1}\left(\cos x\right)=1-\frac{x}{n}$
$1-\frac{x}{n}$ is the equation of straight line, which passes through the fixed point $(0,1)$.
From graph, it is clear that the equation ${\cos }^{-1}\left(\cos x\right)=\frac{n-x}{n}$, has seven roots for all $x\ge 0$, when $n\in (6\pi ,8\pi )$