Question

设x≥1,比较x³与x²-x+1的大小

详细解题过程~

Answer 1

因为 x^2+1＞0所以 x^2+x+1＞x (1)因为 x≥1故 x-1≥0 (2)所以 (1)的两边同乘以 x-1 得 (x-1)(x^2+x+1)≥x(x-1) 即 x^3-1≥x^2-x，移项得 x^3≥x^2-x+1 .

Answer 2

x^3-(x^2-x+1)=x^3-x^2+x-1=x^2*(x-1)+(x-1)=(x-1)(x^2+1)∵x>=1∴x-1>=0,x^2+1>0∴x^3-(x^2-x+1)=(x-1)(x^2+1)>=0∴x^3>=x^2-x+1(仅当x=1时取等)

Answer 3

x^3-(x^2-x+1)=x^3-x^2+x-1=x^2(x-1) + x-1=(x-1)(x^2+1)因为x≥1所以x-1≥0而x^2+1>0所以x^3-(x^2-x+1)=(x-1)(x^2+1)≥0即当x≥1时x^3≥x^2-x+1