Đề: Cho \(\left\{{}\begin{matrix}x,y,z>0\\x+y\le z\end{matrix}\right.\) tìm Min của \(\left(x^2+y^2+z^2\right)\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\right)\) Làm thế này không biết đúng ko
Ta có :A= \(\left(x^2+y^2+z^2\right)\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\right)=3+\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}+\dfrac{z^2}{x^2}+\dfrac{x^2}{z^2}+\dfrac{z^2}{y^2}+\dfrac{y^2}{z^2}\)
=> A \(=3+\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+\left(\dfrac{x^2}{z^2}+\dfrac{z^2}{16x^2}\right)+\left(\dfrac{y^2}{z^2}+\dfrac{z^2}{16y^2}\right)+\dfrac{15}{16}\left(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\right)\)
Áp dụng BĐT Cauchy ta có
\(A\ge3+2+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{15}{16}\left(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\right)=6+\dfrac{15}{16}\left(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\right)\)
Do \(x+y\le z\Rightarrow\dfrac{x}{z}+\dfrac{y}{z}\le1\) ; Đặt \(u=\dfrac{x}{z}\); \(v=\dfrac{y}{z}\)
\(\Rightarrow\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}=\dfrac{1}{u^2}+\dfrac{1}{v^2}\ge\dfrac{2}{uv}\ge\dfrac{2}{\dfrac{\left(u+v\right)^2}{4}}\ge\dfrac{2}{\dfrac{1}{4}}=8\)
\(\Rightarrow A\ge6+\dfrac{15}{16}.8=\dfrac{27}{2}\) Vậy minA = \(\dfrac{27}{2}\) khi \(x=y=\dfrac{z}{2}\)
\(BDT\Leftrightarrow\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}+\dfrac{y^2}{z^2}+\dfrac{z^2}{y^2}+\dfrac{z^2}{x^2}+\dfrac{x^2}{z^2}+3\)
Áp dụng BĐT AM-GM:\(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\ge2\)
\(\Rightarrow VT\ge\)\(\dfrac{y^2}{z^2}+\dfrac{z^2}{y^2}+\dfrac{z^2}{x^2}+\dfrac{x^2}{z^2}+5\)
Lần lượt có các đánh giá: \(\dfrac{y^2}{z^2}+\dfrac{x^2}{z^2}\ge\dfrac{1}{2}\left(\dfrac{x+y}{z}\right)^2\)
Và \(\dfrac{z^2}{y^2}+\dfrac{z^2}{x^2}\ge\dfrac{1}{2}\left(\dfrac{4z}{x+y}\right)^2\)
\(\Rightarrow VT\ge\dfrac{1}{2}\left(\dfrac{4z}{x+y}\right)^2+\dfrac{1}{2}\left(\dfrac{x+y}{z}\right)^2+5\)
Đặt \(t=\dfrac{z}{x+y}\ge1\) thì ta được:
\(\Rightarrow VT\ge8t^2+\dfrac{1}{2t^2}+5\)\(\ge\dfrac{17}{2}+5=\dfrac{27}{2}\)
