Consider f(x)= ^ -1 ( 2 x 9-4 x² )- ^ -1 ((x)/(3)). Identify which...

Consider

f(x)=\tan ^{-1}

\left(\frac{2 x}{\sqrt{9-4 x²}}\right)-\cos ^{-1}

\left(\frac{x}{3}\right).

Identify which of the following statement(s) is(are) correct? Consider f(x)= ^ -1 ( 2 x 9-4 x² )- ^ -1 ((x)/(3)). Identify which of the following statement(s) is(are) correct?

Options

ANumber of solutions of the equation $f(x)=\ln (-x)$ is 2 .
BNumber of solutions of the equation $f(x)=\ln (-x)$ is 1 .
CIf $f(x)-k=0$ has a solution then number of integral values of $k$ is 4 .
DIf $f(x)-k=0$ has a solution then number of integral values of $k$ is 3 .

Correct answer

C. If $f(x)-k=0$ has a solution then number of integral values of $k$ is 4 .

Step-by-step solution

f(x)=\tan ^{-1}

\left(\frac{2 x}{\sqrt{9-4 x^2}}\right)-\cos ^{-1} \left(\frac{x}{3}\right) Domain of

f(x)

is

x \in\left(\frac{-3}{2}, \frac{3}{2}\right)

f(x)=\sin ^{-1} \left(\frac{2 x}{3}\right)-\cos ^{-1}\left(\frac{x}{3}\right)

When,

\begin{aligned} & x \in\left(\frac{-3}{2}, \frac{3}{2}\right) \\ & \left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\left(\frac{2 \pi}{3}, \frac{\pi}{3}\right)\end{aligned}

Range of

f(x)

\left(\frac{-7 \pi}{6}+\frac{\pi}{6}\right) Number of solution of

f(x)=\ln (-x)

has only one solution

f(x)-k=0 \quad

has number of integral solution

k=\{-3,-2,-1,0\} \quad \Rightarrow \quad 4

value$$

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