a.
\(2x+3y+\dfrac{1}{x}+\dfrac{8}{y}=\left(x+\dfrac{1}{x}\right)+2\left(y+\dfrac{4}{y}\right)+\left(x+y\right)\ge2\sqrt{\dfrac{x}{x}}+2.2\sqrt{\dfrac{4y}{y}}+3=13\)
Dấu "=" xảy ra khi \(\left(x;y\right)=\left(1;2\right)\)
b.
\(a\ge3b\Rightarrow\dfrac{a}{b}\ge3\)
\(F=\dfrac{a^4}{ab^3}+\dfrac{b^4}{ab^3}=\dfrac{a^3}{b^3}+\dfrac{b}{a}=\left(\dfrac{a^3}{243b^3}+\dfrac{b}{3a}+\dfrac{b}{3a}+\dfrac{b}{3a}\right)+\dfrac{242}{243}.\left(\dfrac{a}{b}\right)^3\)
\(F\ge4\sqrt[4]{\dfrac{a^3}{243.3^3.b^3}}+\dfrac{242}{243}.3^3=\dfrac{82}{3}\)
Dấu "=" xảy ra khi \(a=3b\)
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