已知x+y=1.求证x4+y4≥1/8

Question

已知x+y=1.求证x4+y4≥1/8

Answer 1

x4+y4≥2√(x^4y^4)=2x²y²
x+y≥2√|xy|
4|xy|≤1
|xy|≤1/4
|xy|²≤1/16
所以
2x²y²≤1/8
即
x^4+y^4≥2x²y²=1/8