Question

Let $g:R\to R$ be defined as $g(x)=\mathrm{sgn}\left(x^2-5x+6\right)$, then find the number of solutions of equation $\sin x={\cos }^{-1}\left(g\left({\sin }^{-1}x\right)\right)$ lying in interval$[0,314].$
[Note: $\mathrm{sgn}(k)$ denotes the signum function of $k$.]

Answer 1

Answer: 1

As, $g\left({\sin }^{-1}x\right)=\mathrm{sgn}\left(\left({\sin }^{-1}x-2\right)\cdot \left({\sin }^{-1}x-3\right)\right)=1$
$\therefore$ We get, $\sin x={\cos }^{-1}(1)\Rightarrow \sin x=0$
$\Rightarrow x=n\pi ,n\in I$
But domain of equation is $[-1,1]$.
$\therefore$ The possible solution is $x=0$.
Hence, the number of solutions of given equation are one.