Question

证明：1+2x^4>=2x^3+x^2

Answer 1

2x^4-2x^3-x^2+1=2x^3(x-1)-(x^2-1)=2x^3(x-1)-(x+1)(x-1)=(x-1)[2x^3-（x+1)]=(x-1)(x-1)(2x^2+2x+1)=(x-1)^2(2x^2+2x+1)=(x-1)^2[x^2+(x+1)^2]≥0,所以1+2x^4≥2x^3+x^2.