Question

How to solve this inequation?`sin^4x+cos^4x>=1/2`

Answer 1

Inequality always true.

Explanation:

`sin^4 x + cos^4 x = (sin^2 x + cos^2 x)^2 - 2sin^2 x.cos^2 x =`
`= 1 - 2sin^2 x.cos^2 x = 1 - (sin^2 2x)/2 >= 1/2`
The inequality becomes:
`(sin^2 (2x))/2 <= 1/2`
`sin^2 (2x) <= 1`
`sin 2x <= +- 1`
Solve this inequality in 2 cases:
a. `sin 2x <= 1` -->
The answer is undefined, because sin 2x always < 1, regardless of
the value of x.
b. `sin 2x <= - 1` (rejected because < - 1)
Conclusion: The inequality is always true.
Check.
`x = pi/6 --> sin^4x + cos^4 x = 1/16 + 9/16 = 10/16 = 5/8 > 1/2`
`x = (2pi)/3 --> sin^4 x + cos^4x = 1/16 + 9/16 = 10/16 = 5/8`