Question

The function f(x) = tan^–1(sin x + cos x) is an increasing function in ______.

Options

• `(0, π/2)`

• `(-π/2, π/2)`

• `(π/4, π/2)`

• `(-π/2, π/4)`

Answer 1

The function f(x) = tan^–1(sin x + cos x) is an increasing function in `underlinebb((-π/2, π/4)`.

Explanation:

Given that f(x) = tan^–1(sin x + cos x)

Differentiate w.r. to x

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

= `(sqrt(2).(1/sqrt(2)cos x - 1/sqrt(2) sin x))/(1 + (sin x + cos x)^2`

= `(sqrt(2)(cos  π/4. cos x - sin  π/4. sin x))/(1 + (sin x + cos x)^2`

∴ f'(x) = `(sqrt(2)cos(x + π/4))/(1 + (sin x + cos x)^2`

Given that f(x) is increasing

∴ f'(x) > 0

`\implies cos(x + π/4) > 0` 

`\implies - π/2 < x + π/4 < π/2`

`\implies - (3π)/4 < x < π/4`

Hence, f(x) is increasing when `n ∈ (-π/2, π/4)`