Question

The function $f\left(x\right)=\tan x–x$

• A. always increases
• B. sometimes increases and sometimes decreases
• C. never increases
• D. always decreases

Answer 1

The correct option is A always increases

Given: $f\left(x\right)=\tan x-x$

Differentiating the above eq. w.r.t $x$,

$\Rightarrow f^{\prime }\left(x\right)={\sec }^{2}x-1$

We know, ${\sec }^{2}x\ge 1,\forall x\in R$

$\therefore f^{\prime }\left(x\right)\ge 0,\forall x\in R$

$\therefore f\left(x\right)$ is always increasing for all real $x$.

Hence, option (a) is correct.