a/b² + b/a² - 1/a - 1/b =
(a³ + b³ - ab² - a²b) / (a²b²) =
(a³ + b³ + 3ab² + 3a²b - 4ab² - 4a²b) / (a²b²) =
((a + b)³ - 4ab(a + b)) / (a²b²) =
( (a + b)( (a + b)² - 4ab ) / (a²b²) =
(a + b)(a - b)² / (a²b²) >= 0, т.к. (a - b)² >= 0, a²b² > 0, a + b > 0 при a > 0, b > 0
значит
a/b² + b/a² - 1/a - 1/b >= 0,
a/b² + b/a² >= 1/a + 1/b
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Answers & Comments
a/b² + b/a² - 1/a - 1/b =
(a³ + b³ - ab² - a²b) / (a²b²) =
(a³ + b³ + 3ab² + 3a²b - 4ab² - 4a²b) / (a²b²) =
((a + b)³ - 4ab(a + b)) / (a²b²) =
( (a + b)( (a + b)² - 4ab ) / (a²b²) =
(a + b)(a - b)² / (a²b²) >= 0, т.к. (a - b)² >= 0, a²b² > 0, a + b > 0 при a > 0, b > 0
значит
a/b² + b/a² - 1/a - 1/b >= 0,
a/b² + b/a² >= 1/a + 1/b