Question

Find the sub intervals in which f(x) = cot^–1 (sin x + cos x), x ∈ (0, π) is increasing and decreasing.

Answer 1

Given,

f(x) = cot^–1 (sin x + cos x)

Differential both sides w.r.t ‘x’, we get

f'(x) = `(-1)/(1+(sinx+cosx)^2)*(cosx-sinx)`

f'(x) = `(sinx-cosx)/(1+(sinx+cosx)^2)`

f'(x) = `(sinx-cosx)/(2+sin2x)`

If f(x) is increasing, f'(x) = > 0

sinx – cosx > 0

⇒ sinx > cosx

If f(x) is decreasing, f'(x) < 0

sinx – cosx < 0 

⇒ sinx < cosx

If f'(x) = 0, sinx = cosx; x = `pi/4` 

In interval `(0,pi/4)`; cosx > sinx;

Hence, f(x) is decreasing in `(0,pi/4)`.

In interval `(pi/4,pi)`; sinx > cosx; 

Hence, f(x) is increasing in `(pi/4,pi)`.