Question

Function sin^4 x+ cos^4 x is incresing in which quadrant

Answer 1

1st quadrant

HII ! MATES

Answer 2

Answer:

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It is not increasing throughout any quadrant.

For x ∈ [ 0, 2π ], this function is increasing on the intervals:

( π/4, π/2 ),  ( 3π/4, π ),  ( 5π/4, 3π/2 ) and ( 7π/4, 2π )

Step-by-step explanation:

Things we need to know are:

• sin² x + cos² x = 1
• sin 2x = 2 sin x cos x
• cos 2x = 1 - 2 sin² x

Then

sin⁴ x + cos⁴ x

= ( sin² x + cos² x )² - 2 sin² x cos² x

= 1 - (1/2) ( 2 sin x cos x ) ( 2 sin x cos x )

= 1 - (1/2) sin² 2x

= 1 - 1/4 + (1/4) ( 1 - 2 sin² 2x )

= 3/4 + (1/4) cos 4x

So the function is increasing precisely when cos 4x is increasing.

We know cos x increases on the intervals

( π, 2π ),  ( 3π, 4π ),  ( 5π, 6π ),  ( 7π, 8π )

The function cos 4x is just cos x compressed in towards the y-axis by a scale of 4.  So cos 4x is increasing on the intervals

( π/4, π/2 ),  ( 3π/4, π ),  ( 5π/4, 3π/2 ) and ( 7π/4, 2π )