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

1 Answer
Nghi N · Nghi N.
Mar 22, 2017

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

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