設(shè)a,b∈正實(shí)數(shù),且a+b=1,求證:大于等于25/4

數(shù)學(xué)人氣：798 ℃時(shí)間：2020-08-31 05:34:12

優(yōu)質(zhì)解答

應(yīng)該是(a+1/a)(b+1/b)≥25/4
（a+1/a)(b+1/b)
=（a²+1)(b²+1)/(ab)
=(a²+b²+a²b²+1)/(ab)
=[(a²+b²+2ab)-2ab+a²b²+1]/(ab)
=[(a+b)²+a²b²-2ab+1]/(ab) 【a+b=1]
=(a²b²-2ab+2)/(ab)
=ab+2/(ab)-2
∵a+b=1,a>0,b>0
a+b≥2√(ab)
∴ab≤1/4
又將ab看成自變量,函數(shù)ab+2/(ab)是減函數(shù)
ab=1/4時(shí),ab+2/(ab)取得最小值1/4+8=33/4
∴ab+2/(ab)-2≥33/4-2=25/4
即是(a+1/a)(b+1/b)≥25/4