Question

The function \( f(x) = \tan x - x \)

• A. always decreases
• B. never increases
• C. neither increases nor decreases
• D. always increases

Hint:

Whenever you have a function involving trigonometric functions and algebraic terms, differentiate it to check whether the function is increasing or decreasing. The square of a tangent function will always be non-negative, implying that the function increases.

Answer 1

The Correct Option is D

We are given: \[ f(x) = \tan x - x \] Now, differentiate \( f(x) \): \[ f'(x) = \sec^2 x - 1 = \tan^2 x \] Since \( \tan^2 x \geq 0 \) for all \( x \), the function is always increasing for the given domain. Thus, the correct answer is option (4), which states that the function always increases.