Question

f(x) = sin⁴ x + cos⁴ x, x ∈(0, π/4)

Answer 1

f(x) = sin⁴ x + cos⁴ x

Diff. w.r.t x,

f'(x) = 4 sin³ x cos x + 4 cos³ x(-sin x)

⇒ f'(x) = 4 sin x cos x (sin² x – cos² x)

⇒ f'(x) = -2.2 sin x cos x (cos² x – sin² x)

⇒ f'(x) = – 2sin 2x cos 2x

⇒ f'(x) = – sin 4x

In interval (0, π/4), - sin4x < 0

or f'(x) < 0

Hence, function is decreasing in internal (0, π/4)