高二数学题?

设A=1+2x4,B=2x3+x2,x属于Ｒ，则Ａ,B的大小关系为＿＿＿＿？

热心网友

作差法:A-B=1+2x^4-2x^3-x^2 =2(x^4-x^3)+1-x^2 =2x^3(x-1)+(1+x)(1-x) =(x-1)(2x^3-x-1) =(x-1)(2x^3-2x^2+2x^2-x-1) =(x-1)[2x^2(x-1)+(x-1)(2x+1)]=(x-1)^2(2x^2+2x+1)∵2x^2+2x+1=2[x+(1/2)]^2+1/2＞0 ∴上式≥0即A≥B

热心网友

A-B=(1-x^2)+2x^3(x-1)=2x^3(x-1)-(x-1)(x+1)=(x-1)(2x^3-x-1)=(x-1)[(x^3-x)+(x^3-1)]=(x-1)[x(x+1)(x-1)+(x-1)(x^2+x+1)]=(x-1)[(x-1)(x^2+x+x^2+x+1)]=(x-1)^2(2x^2+2x+1)=(x-1)^2[(√2x+1/√2)^2+1/2]所以A-B≥0,所以A≥B

热心网友

A-B=1+2x^4-2x^3-x^2 =2(x^4-x^3)+1-x^2 =2x^3(x-1)+(1+x)(1-x) =(x-1)(2x^3-x-1) =(x-1)(2x^3-2x^2+2x^2-x-1) =(x-1)[2x^2(x-1)+(x-1)(2x+1)]=(x-1)^2(2x^2+2x+1)∵2x^2+2x+1=2[x+(1/2)]^2+1/2＞0 ∴上式≥0即A≥B