Question

The function f(x) = sin⁴x + cos⁴x is an increasing function if ______.

Options

• `π/4 < x < π/2`

• `0 < x < π/4`

• `π/8 < x < π/4`

• `0 < x < π/8`

Answer 1

The function f(x) = sin⁴x + cos⁴x is an increasing function if `underlinebb(π/4 < x < π/2)`.

Explanation:

f(x) = sin⁴x + cos⁴x

∴ f'(x) = 4sin³x cosx + 4cos³x (– sinx)

= 4sinx cosx (sin²x – cos²x)

= –2·2sinx cosx (cos²x – sin²x)

= –2·sin2x·cos2x

= – sin 4x

f(x) is an increasing function if

f'(x) > 0,

i.e. –sin 4x > 0,

i.e. sin 4x < 0,

i.e. π < 4x < 2π

i.e. `π/4 < x < π/2`