cho a,b>0,tìm min P= \(P=\frac{a+b}{\sqrt{ab}}+\frac{\sqrt{ab}}{a+b}\)
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HL
1
17 tháng 7 2019
\(A=\left(\frac{a+b}{\sqrt{ab}}+\frac{4\sqrt{ab}}{a+b}\right)-\frac{3\sqrt{ab}}{a+b}\ge2\sqrt{\frac{4\sqrt{ab}\left(a+b\right)}{\sqrt{ab}\left(a+b\right)}}-\frac{3\sqrt{ab}}{2\sqrt{ab}}=4-\frac{3}{2}=\frac{5}{2}\)
Dấu "=" xảy ra \(\Leftrightarrow\)\(\hept{\begin{cases}\frac{a+b}{\sqrt{ab}}=\frac{4\sqrt{ab}}{a+b}\\\left(a+b\right)^2=4ab\end{cases}\Leftrightarrow a=b}\)
KK
2
13 tháng 7 2018
Đặt \(\frac{a+b}{\sqrt{ab}}=t\ge2\)
Thế vào :\(A\ge\frac{\sqrt{ab}}{a+b}+\frac{16.\frac{\left(a+b\right)^2}{2}}{ab}=\frac{\sqrt{ab}}{a+b}+\frac{8\left(a+b\right)^2}{ab}=\frac{1}{t}+8t^2\)
\(=\frac{1}{2t}+\frac{1}{2t}+\frac{1}{16}t^2+\frac{127t^2}{16}\)
\(\ge\sqrt[3]{\frac{1}{2t}.\frac{1}{2t}.\frac{t^2}{16}}+\frac{127t^2}{16}=3\sqrt[3]{\frac{1}{4}.\frac{1}{16}}+\frac{127t^2}{16}\ge\frac{3}{4}+\frac{127.2^2}{16}=\frac{3}{4}+\frac{127}{4}=\frac{130}{4}=\frac{65}{2}\)
Vậy min A=\(\frac{65}{2}\) đạt được khi \(t=2\Rightarrow\frac{a+b}{\sqrt{ab}}=2\Rightarrow\left(\sqrt{a}-\sqrt{b}\right)^2=0\Rightarrow a=b\)
16 tháng 7 2018
sorry,hàng thứ 4 biểu thức đầu tiên là \(3\sqrt[3]{\frac{1}{2t}.\frac{1}{2t}.\frac{t^2}{16}}\) nha
4
2
30 tháng 12 2018
\(A=\frac{a}{\sqrt{3+a^2}}+\frac{b}{\sqrt{3+b^2}}+\frac{c}{\sqrt{3+c^2}}\)
\(=\frac{a}{\sqrt{a^2+ab+bc+ca}}+\frac{b}{\sqrt{b^2+bc+ca+ab}}+\frac{c}{\sqrt{c^2+ca+ab+bc}}\)
\(=\frac{\sqrt{a}\cdot\sqrt{a}}{\sqrt{\left(a+b\right)\left(a+c\right)}}+\frac{\sqrt{b}\cdot\sqrt{b}}{\sqrt{\left(b+c\right)\left(a+b\right)}}+\frac{\sqrt{c}\cdot\sqrt{c}}{\sqrt{\left(c+a\right)\left(c+b\right)}}\)
\(=\frac{\sqrt{a}}{\sqrt{a+b}}\cdot\frac{\sqrt{a}}{\sqrt{c+a}}+\frac{\sqrt{b}}{\sqrt{b+c}}\cdot\frac{\sqrt{b}}{\sqrt{a+b}}+\frac{\sqrt{c}}{\sqrt{c+a}}\cdot\frac{\sqrt{c}}{\sqrt{c+b}}\)
\(\le\frac{\frac{a}{a+b}+\frac{a}{c+a}}{2}+\frac{\frac{b}{b+c}+\frac{b}{a+b}}{2}+\frac{\frac{c}{c+a}+\frac{c}{b+c}}{2}\)
\(=\frac{\frac{a+b}{a+b}+\frac{b+c}{b+c}+\frac{c+a}{c+a}}{2}=\frac{3}{2}\)
Vậy Max A = 3/2 khi a = b = c = 1. (Max not Min)
13 tháng 5 2021
Đặt \(\left(x;y;z\right)=\left(\frac{1}{a};\frac{1}{b};\frac{1}{c}\right)\) \(\left(x,y,z>0\right)\)
Theo đề \(ab+bc+ca=3abc\Leftrightarrow\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=\frac{3}{xyz}\)
\(\Rightarrow x+y+z=3\)
Và \(\sqrt{\frac{ab}{a+b+1}}+\sqrt{\frac{bc}{b+c+1}}+\sqrt{\frac{ca}{c+a+1}}\)
\(=\sqrt{\frac{\frac{1}{xy}}{\frac{1}{x}+\frac{1}{y}+1}}+\sqrt{\frac{\frac{1}{yz}}{\frac{1}{y}+\frac{1}{z}+1}}+\sqrt{\frac{\frac{1}{zx}}{\frac{1}{z}+\frac{1}{x}+1}}\)
\(=\frac{1}{\sqrt{x+y+xy}}+\frac{1}{\sqrt{y+z+yz}}+\frac{1}{\sqrt{z+x+zx}}\)
\(\ge\frac{9}{\sqrt{x+y+xy}+\sqrt{y+z+yz}+\sqrt{z+x+zx}}\) (Cauchy Schwarz)
Ta có: \(\sqrt{x+y+xy}+\sqrt{y+z+yz}+\sqrt{z+x+zx}\)
\(=\sqrt{\left(\sqrt{x+y+xy}+\sqrt{y+z+yz}+\sqrt{z+x+zx}\right)^2}\)
\(\le\sqrt{3\left(x+y+xy+y+z+yz+z+x+zx\right)}\)
\(=\sqrt{\left[2\left(x+y+z\right)+\left(xy+yz+zx\right)\right]}\)
\(\le\sqrt{6+\frac{\left(x+y+z\right)^2}{3}}=\sqrt{6+\frac{3^2}{3}}=3\)
\(\Rightarrow\sqrt{\frac{ab}{a+b+1}}+\sqrt{\frac{bc}{b+c+1}}+\sqrt{\frac{ca}{c+a+1}}\)
\(\ge\frac{9}{\sqrt{x+y+xy}+\sqrt{y+z+yz}+\sqrt{z+x+zx}}\ge\frac{9}{3}=3\)
Dấu "=" xảy ra khi: \(x=y=z=1\Rightarrow a=b=c=1\)