Áp dụng BĐT Cô-si cho 2 số thực dương \(\dfrac{xy}{z}\) và \(\dfrac{yz}{x}\) có:
\(\dfrac{xy}{z}+\dfrac{yz}{x}\) \(\ge\) 2\(\sqrt{\dfrac{xy}{z}\cdot\dfrac{yz}{x}}\) = 2\(\sqrt{y^2}\) = 2y (1)
Tương tự: \(\dfrac{yz}{x}+\dfrac{zx}{y}\ge2z\) (2)
\(\dfrac{xy}{z}+\dfrac{zx}{y}\ge2x\) (3)
Từ (1); (2); (3)
\(\Rightarrow\) \(\dfrac{2xy}{z}+\dfrac{2yz}{x}+\dfrac{2zx}{y}\ge2x+2y+2z\)
\(\Leftrightarrow\) 2\(\left(\dfrac{xy}{z}+\dfrac{yz}{x}+\dfrac{zx}{y}\right)\) \(\ge\) 2(x + y + z)
\(\Leftrightarrow\) \(\dfrac{xy}{z}+\dfrac{yz}{x}+\dfrac{zx}{y}\ge x+y+z=10\)
Hay PMin = 10
Dấu "=" xảy ra \(\Leftrightarrow\) x = y = z = \(\dfrac{10}{3}\)
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