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Bất đẳng thức giữa trung bình cộng và trung bình nhân (AG - GM) là gì? Bất đẳng thức Cauchy là gì? - Toán 9

1. Bất đẳng thức giữa trung bình cộng và trung bình nhân (AG - GM) (Bất đẳng thức Cauchy)

Cho hai số \(a,b \ge 0\). Trung bình cộng của hai số là \(\frac{{a + b}}{2}\); Trung bình nhân của hai số là \(\sqrt {ab} \).

Khi đó, ta luôn có: \(\frac{{a + b}}{2} \ge \sqrt {ab} \) (1)

Bất đẳng thức (1) được gọi là bất đẳng thức giữa trung bình cộng và trung bình nhân cho hai số không âm a và b (thường viết tắt là AM – GM). Ta cũng thường gọi bất đẳng thức trên là bất đẳng thức Cauchy.

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