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\(P=\frac{2}{a^2+b^2}+\frac{2}{2ab}+\frac{34}{ab}+\frac{17ab}{8}-\frac{ab}{8}\)
\(P=2\left(\frac{1}{a^2+b^2}+\frac{1}{2ab}\right)+\frac{34}{ab}+\frac{17ab}{8}-\frac{ab}{8}\)
\(P\ge2\cdot\frac{4}{a^2+b^2+2ab}+2\sqrt{\frac{34}{ab}\cdot\frac{17ab}{8}}-\frac{\frac{\left(a+b\right)^2}{4}}{8}\)
( do \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y};x+y\ge2\sqrt{xy};ab\le\frac{\left(a+b\right)^2}{4}\))
\(\Rightarrow P\ge\frac{8}{\left(a+b\right)^2}+2\sqrt{\frac{289}{4}}-\frac{\frac{4^2}{4}}{8}\)
\(\Rightarrow P\ge\frac{8}{16}+17-\frac{1}{2}=17\)
\(P=17\Leftrightarrow\left\{{}\begin{matrix}a^2+b^2=2ab\\\frac{34}{ab}=\frac{17ab}{8}\\a=b\\a+b=4\end{matrix}\right.\Leftrightarrow a=b=2\)
Vậy Min P = 17 \(\Leftrightarrow a=b=2\)
\(A=\frac{2}{a^2+b^2}+\frac{35}{ab}+2ab\)
\(=2\left(\frac{1}{a^2+b^2}+\frac{1}{2ab}\right)+\frac{34}{ab}+\frac{17}{8}ab-\frac{1}{8}ab\)
\(\ge2.\frac{4}{a^2+b^2+2ab}+2\sqrt{\frac{34}{ab}.\frac{17}{8}ab}-\frac{1}{8}.\frac{\left(a+b\right)^2}{4}\)
\(\Leftrightarrow A\ge2.\frac{4}{\left(a+b\right)^2}+2.\frac{17}{2}-\frac{1}{8}.\frac{4}{4^2}+17-\frac{1}{2}\)
\(\Leftrightarrow A\ge\frac{1}{2}+17-\frac{1}{2}=17\)
Dấu " = " xảy ra \(\Leftrightarrow a=b=2\)
Chúc bạn học tốt !!!
Ta có : \(4\ge a+b\ge2\sqrt{ab}\Rightarrow ab\le4\)
Áp dụng bất đẳng thức \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\)(bạn có thể chứng minh bằng biến đổi tương đương)
Ta có :\(P=\frac{2}{a^2+b^2}+\frac{35}{ab}+2ab=\left(\frac{2}{a^2+b^2}+\frac{1}{ab}\right)+\left(\frac{32}{ab}+2ab\right)+\frac{2}{ab}=2\left(\frac{1}{a^2+b^2}+\frac{1}{2ab}\right)+\left(\frac{32}{ab}+2ab\right)+\frac{2}{ab}\ge\frac{2.4}{\left(a+b\right)^2}+2\sqrt{\frac{32}{ab}.2ab}+\frac{2}{ab}\ge\frac{8}{4^2}+2.8+\frac{2}{4}=17\)Dấu đẳng thức xảy ra \(\Leftrightarrow\hept{\begin{cases}a=b\\a^2b^2=16\\0< a+b\le4\end{cases}\Leftrightarrow}a=b=2\)
Vậy \(MinP=17\Leftrightarrow a=b=2\)
\(a+b\ge2\sqrt{ab}\Leftrightarrow2\sqrt{ab}\le4\Leftrightarrow ab\le4\)
\(P=\left(\dfrac{2}{a^2+b^2}+\dfrac{1}{ab}\right)+\dfrac{2}{ab}+2ab+\dfrac{32}{ab}\\ \Leftrightarrow P=2\left(\dfrac{1}{a^2+b^2}+\dfrac{1}{2ab}\right)+\dfrac{2}{ab}+2ab+\dfrac{32}{ab}\\ \Leftrightarrow P\ge2\cdot\dfrac{4}{a^2+b^2+2ab}+2\sqrt{\dfrac{32}{ab}\cdot2ab}+\dfrac{2}{4}\\ \Leftrightarrow P\ge\dfrac{8}{\left(a+b\right)^2}+2\sqrt{64}+\dfrac{1}{2}\\ \Leftrightarrow P\ge\dfrac{8}{16}+16+\dfrac{1}{2}=17\)
Dấu \("="\Leftrightarrow a=b=2\)
\(\frac{1}{a^2+b^2}+\frac{1}{2ab}\ge\frac{4}{\left(a+b\right)^2}\ge\frac{1}{4}\Rightarrow\frac{2}{a^2+b^2}+\frac{1}{ab}\ge\frac{1}{2}\)
\(\frac{32}{ab}+2ab\ge2\sqrt{64}=16\)(cô-si)
tự xét nốt 2/ab nhé
