Chứng minh bất đẳng thức sau
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VD1.
Cho \(\{a_1, a_2, \ldots, a_n \in \mathbb{R}\}\)
\(\{x_1, x_2, \ldots, x_n \in \mathbb{R}\}\)
CMR: \((a_1^2 + a_2^2 + \ldots + a_n^2)(x_1^2 + x_2^2 + \ldots + x_n^2) \geq (a_1 x_1 + a_2 x_2 + \ldots + a_n x_n)^2\)
Dấu bằng xảy ra khi \(\frac{a_1}{x_1} = \frac{a_2}{x_2} = \ldots = \frac{a_n}{x_n}\)
VD2. CMR: \((a^2 + b^2)(b^2 + c^2)(c^2 + a^2) \geq 8a^2b^2c^2 \quad \forall a, b, c.\) Dấu bằng khi nào
VD3. Cho \(\{a, b, c, d > 0\}\)
\(\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c} \geq \frac{3}{1 + d} \quad \text{CMR: } abc d \leq \frac{1}{81}\) Dấu bằng khi nào
VD4. CMR:
\(\frac{1 + a + b + c}{3} \geq (1 + a)(1 + b)(1 + c) \geq (1 + \sqrt[3]{abc})^3 \geq 8\sqrt{abc} \quad \forall a, b, c \geq 0\)
VD1.
Cho \(\{a_1, a_2, \ldots, a_n \in \mathbb{R}\}\)
\(\{x_1, x_2, \ldots, x_n \in \mathbb{R}\}\)
CMR: \((a_1^2 + a_2^2 + \ldots + a_n^2)(x_1^2 + x_2^2 + \ldots + x_n^2) \geq (a_1 x_1 + a_2 x_2 + \ldots + a_n x_n)^2\)
Dấu bằng xảy ra khi \(\frac{a_1}{x_1} = \frac{a_2}{x_2} = \ldots = \frac{a_n}{x_n}\)
VD2. CMR: \((a^2 + b^2)(b^2 + c^2)(c^2 + a^2) \geq 8a^2b^2c^2 \quad \forall a, b, c.\) Dấu bằng khi nào
VD3. Cho \(\{a, b, c, d > 0\}\)
\(\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c} \geq \frac{3}{1 + d} \quad \text{CMR: } abc d \leq \frac{1}{81}\) Dấu bằng khi nào
VD4. CMR:
\(\frac{1 + a + b + c}{3} \geq (1 + a)(1 + b)(1 + c) \geq (1 + \sqrt[3]{abc})^3 \geq 8\sqrt{abc} \quad \forall a, b, c \geq 0\)