Question

若x大于0且x≠1,则1+x^7与x^3+x^4的大小关系

Answer 1

1+x^7-(x^3+x^4)

=1-x^3+x^7-x^4

=(1-x^3)-x^4(1-x^3)

=(1-x^4)(1-x^3)

=(x^4-1)(x^3-1)

=(x+1)(x-1)(x^2+1)(x-1)(x^2+x+1)

=(x^2+1)(x-1)^2(x+1)(x^2+x+1)

x^2+1>0

x≠1,(x-1)^2>0

x^2+x+1=(x+1/2)^2+3/4>0

x>0,x+1>0

所以(x^2+1)(x-1)^2(x+1)(x^2+x+1)>0

所以1+x^7-(x^3+x^4)>0

1+x^7>x^3+x^4