Question

试说明不管ab取何值2a^2+5b^2-2ab+2a-4b+5的值始终为正数.

Answer 1

2a²+5b²-2ab+2a-4b+5

=(a²-2ab+b²)+(a²+2a+1)+(4b²-4b+1)+3

=(a-b)²+(a+1)²+(2b-1)²+3

∵(a-b)²≥0,(a+1)²≥0,(2b-1)²≥0

∴(a-b)²+(a+1)²+(2b-1)²+3≥3

∴2a²+5b²-2ab+2a-4b+5≥3

即：不管ab取何值2a²+5b²-2ab+2a-4b+5的值始终为正数.