已知a>0,b>0,(1)求证a+1⼀a>=2;(2)b⼀a + a⼀b >=2

2025-05-13 02:06:33
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回答(1):

(1)a+1/a=(a²+1)/a=(a²+1-2a)/a+2=(a-1)²/a+2
(a-1)²≥0 a>0
所以(a-1)²/a+2≥2
a+1/a>=2
(2)
b/a + a/b
=(b²+a²)/ab
=(b²+a²-2ab)/ab+2
=(a-b)²/ab+2
a>0,b>0 ab>0 (a-b)²≥0
所以(a-b)²/ab+2≥2
b/a + a/b >=2