Question

Show that f(x) = tan^–1(sin x + cos x) is an increasing function in (0, π/4).

Answer 1

Given: f (x) = tan^–1(sin X + cos X)

To show: the given function is increasing in 0, π/4 .

Explanation: Given f (x) = tan^–1(sin X + cos X)

Applying first derivative with respect to x, we get

Expanding (sin x+cos x )², we get

But sin²x+cos²x = 1 and 2sin Xcos X = sin2x, so the above equation becomes,

Now for f(x) to be decreasing function,

f’(x)≥0