Question

The function f(x) = tan x - x
(a) always increases (b) always decreases
(c) never increases (d) sometimes increases sometime decreases

Answer 1

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

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

Differentiating both sides with respect to x, we get

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

We know

$\sec x\in (-\infty ,-1]\cup [1,\infty )$

$\Rightarrow {\sec }^{2}x\in [1,\infty )$

Or sec²x ≥ 1 for all real values of x

$\therefore f\text{'}\left(x\right)\ge 0\forall x\in \mathrm{R}$

⇒ f(x) is increasing for all x ∈ R

Thus, the function f(x) always increases for all real values of x.

Hence, the correct answer is option (a).