Show that the function f(x) = cot^{–1} (sin x + cos x) is decreasing on (0, π/4) and increasing on (π/4, π/2).

Given:- Function f(x) = cot^{–1} (sin x + cos x)

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

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

(ii) If f’(x) < 0 for all , 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,

f(x) = cot^{–1} (sin x + cos x)

⇒

Now, as given

⇒ Cosx – sinx< 0 ; as here cosine values are smaller than sine values for same angle

⇒

⇒ f’(x) < 0

hence, Condition for f(x) to be decreasing

Thus f(x) is decreasing on interval

17