bất đẳng thức ( cân bằng hệ số với AM-GM)

T

thaolovely1412

Ta có: [TEX]P=3x+2y+z\geq 6\sqrt[6]{x^{3}y^{2}z}[/TEX]
mà [TEX]2xyz=3x^{2}+4y^{2}+5z^{2}\geq 12\sqrt[12]{(x^{2})^{3}(y^{2})^{4}(z^{2})^{5}}=12\sqrt{x}. \sqrt[6]{z^{5}}.\sqrt[3]{y^{2}}\Rightarrow \sqrt{x}.\sqrt[3]{y}.\sqrt[6]{z}\geq 6\Rightarrow x^{3}y^{2}z\geq 46656[/TEX]
[TEX]\Rightarrow P \geq 6.\sqrt[6]{46656}=36[/TEX]
Vậy [TEX]MinP=36[/TEX] khi [TEX]x=y=6[/TEX]
 
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