Bài 1 :
\(P=2x+y+\frac{30}{x}+\frac{5}{y}\)
\(=\frac{10x}{5}+\frac{5y}{5}+\frac{30}{x}+\frac{5}{y}\)
\(=\frac{6x}{5}+\frac{4x}{5}+\frac{y}{5}+\frac{4y}{5}+\frac{30}{x}+\frac{5}{y}\)
\(=\left(\frac{6x}{5}+\frac{30}{x}\right)+\left(\frac{4x}{5}+\frac{4y}{5}\right)+\left(\frac{y}{5}+\frac{5}{y}\right)\)
Áp dụng bất đẳng thức Cô - si cho 2 số không âm
\(\frac{6x}{5}+\frac{30}{x}\ge2\sqrt{\frac{6x}{5}.\frac{30}{x}}=2\sqrt{36}=2.6=12\left(1\right)\)
\(\frac{y}{5}+\frac{5}{y}\ge2\sqrt{\frac{y}{5}.\frac{5}{y}}=2\left(2\right)\)
Theo đề bài ta có : \(x+y\ge10\) suy ra
\(\frac{4x}{5}+\frac{4y}{5}=\frac{4\left(x+y\right)}{5}\ge\frac{4.10}{5}=8\left(3\right)\)
Cộng (1) ; (2) và (3) vế với vế ta được :
\(\frac{6x}{5}+\frac{30}{x}+\frac{y}{5}+\frac{5}{y}+\frac{4x}{5}+\frac{4y}{5}\ge12+2+8=22\)
Dấu " = " xay ra \(\Leftrightarrow\left\{{}\begin{matrix}\frac{6x}{5}=\frac{30}{x}\\\frac{y}{5}=\frac{5}{y}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x^2=25\\y^2=25\end{matrix}\right.\)
Vì x ; y dương nên \(\left(x;y\right)=\left(5;5\right)\)
Bài 2 :
Đặt \(x=a+b=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}\)
\(\Leftrightarrow x^3=\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)
\(\Leftrightarrow x^3=2+\sqrt{5}+2-\sqrt{5}+\sqrt[3]{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}.x\)
\(\Leftrightarrow x^3=4+\sqrt[3]{4-5}.x\)
\(\Leftrightarrow x^3=4-3x\)
\(\Leftrightarrow x^3+3x-4=0\)
\(\Leftrightarrow x^3-x^2+x^2-x+4x-4=0\)
\(\Leftrightarrow x^2\left(x-1\right)+x\left(x-1\right)+4\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x^2+x+4\right)=0\)
Vì \(x^2+x+4=x^2+2.x.\frac{1}{2}+\frac{1}{4}+\frac{15}{4}=\left(x+\frac{1}{2}\right)^2+\frac{15}{4}>0\left(\forall x\right)\)
Nên \(x-1=0\Leftrightarrow x=1\)
Vậy \(x=a+b=1\)
\(\Rightarrow\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1\left(đpcm\right)\)
Chúc bạn học tốt !!
