Question

In which interval f(x) = sin⁴x + cos⁴x, x ∈ [0, (π/2)] is increasing function

(a) |(π/4), (π/2)| 

(b) |0, (π/4)| 

(c) |0, (π/2)| 

(d) |(π/4), {(– π)/2}|

Answer 1

The correct option (a) |(π/4), (π/2)|   

Explanation:

f(x) = sin⁴ x + cos⁴ x

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

= 4 sin x ∙ cos x [sin² x – cos² x]

∴ f'(x) = – 2 sin 2x ∙ cos 2x

∴ f'(x) = – sin 4x

we have, x ∈ [0, (π/2)]

∴ 4x ∈ [0, 2π]

Now sin 4x ≤ 0 for x ∈ [(π/4), (π/2)]

∴ f'(x) > 0 for (π/4) < x < (π/2)

∴ f(x) is increasing on [(π/4), (π/2)]