Question

The function $f\left(x\right)=\tan \left(x\right)-x$ is:

• A. Always increasing
• B. Always decreasing
• C. Never decreasing
• D. Sometimes decreasing and sometime decreasing

Answer 1

The correct option is A

Always increasing

Find the nature of interval of the given function

We know that a function $f\left(x\right)$ is increasing then $f\text{'}\left(x\right)>0$

$$\begin{gathered} f\left(x\right)=\tan \left(x\right)-x \\ \Rightarrow y=\tan \left(x\right)-x\left[\mathrm{Let}y=f\left(x\right)\right] \\ \frac{dy}{dx}>0 \\ \Rightarrow {\sec }^2\left(x\right)-1>0 \\ \Rightarrow {\tan }^2\left(x\right)>0 \end{gathered}$$

So, as $f\text{'}\left(x\right)>0$, the function $f\left(x\right)=\tan \left(x\right)-x$ is always increasing.

Hence option $\left(\mathrm{A}\right)$ is correct.