Question

Show that f(x) = tan x is an increasing function on (–π/2, π/2).

Answer 1

Given:- Function f(x) = tan x

Theorem:- Let f be a differentiable real function defined on an open interval (a, b).

(i) If f’(x) > 0 for all x ∈ (a, b), then f(x) is increasing on (a, b)

(ii) If f’(x) < 0 for all x ∈ (a, b), then f(x) is decreasing on (a, b)

Algorithm:-

(i) Obtain the function and put it equal to f(x)

(ii) Find f’(x)

(iii) Put f’(x) > 0 and solve this inequation.

For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.

Here we have,

Now, as given

That is 4^th quadrant, where

⇒ sec²x > 0

⇒ f’(x) > 0

hence, Condition for f(x) to be increasing

Thus f(x) is increasing on interval \(\big(-\frac{\pi}{2},\frac{\pi}{2}\big)\)