Question

In which interval f(x) = tan^–1(sinx + cosx) is strictly increasing?

(a) [{(– π)/2}, 0] 

(b) [0, (π/4)] 

(c) [{(– π)/2}, (π/2)] 

(d) (– π, π)

Answer 1

The correct option (b) [0, (π/4)]

Explanation:

 f(x) = tan^–1(sin x + cos x)

 f'(x) = [1/{1 + (sin x + cos x)²}] (cos x – sin x)

∴ f'(x) = [(cos x – sinx)/(2 + 2 sinx cosx)] = [{cosx (1 – tanx)}/{2 + sin2x}]

Now 2 + sin2x > 0 for all x in [0, (π/4)] also cosx > 0 for all x in [0, (π/4)]

nd 1 – tanx > 0 for all x in [0, (π/4)]

∴ f'(x) > 0

 ∴ f(x) is strictly increasing in [0, (π/4)]