由已知,a+b=1,a>0,b>0
1=a+b≥2√(ab)
√(ab)≤1/2
ab≤1/4
a³+b³=(a+b)(a²-ab+b²)
=a²-ab+b²
=a²-ab+b²+2ab-2ab
=(a+b)²-3ab
=1-3ab
由ab≤1/4,得3ab≤3/4,-3ab≥-3/4,1-3ab≥1-3/4=1/4
所以a³+b³≥1/4