求证:2x^4-x^2大于等于2x^3-1

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/15 21:48:23

求证:2x^4-x^2大于等于2x^3-1
求证:2x^4-x^2大于等于2x^3-1

求证:2x^4-x^2大于等于2x^3-1
(2x^4-x^2)-(2x^3-1)
=2x^4-2x^3-x^2+1
=2x^3(x-1)-(x-1)(x+1)
=(x-1)(2x^3-x-1)
=(x-1)^2(2x^2+2x+1)
=(x-1)^2(x^2+(x+1)^2)
>=0

2x^4-x^2-(2x^3-1)
=2x^4-2x^3-x^2+1
=2x^2(x^2-1)-(x^2-1)
=(x^2-1)(2x^2-1)>=0
则x^2>=1或0<=x^2<=1/2
x>=1或x<=-1,-1/2根号2<=X<=1/2根号2

所以当x>=1或x<=-1,或-1/2根号2<=X<=1/2根号2的时候2x^4-x^2大于等于2x^3-1