Cho abc=1 ,a,b,c>0
CMR \(\frac{1}{a^3\left(b+c\right)}+\frac{1}{b^3\left(a+c\right)}+\frac{1}{c^3\left(a+b\right)}\ge\frac{3}{2}\)
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A B C H
\(cosC=cos30^0=\frac{AC}{BC}=\frac{\sqrt{3}}{2}\)
\(\Rightarrow\)\(BC=\frac{AC.2}{\sqrt{3}}=\frac{16}{\sqrt{3}}\)
\(tanC=tan30^0=\frac{AB}{AC}=\frac{1}{\sqrt{3}}\)
\(\Rightarrow\)\(AB=\frac{AC}{\sqrt{3}}=\frac{8}{\sqrt{3}}\)
\(sinC=sin30^0=\frac{AH}{AC}=\frac{1}{2}\)
\(\Rightarrow\)\(AH=\frac{AC}{2}=4\)
\(Q=\frac{x+y-2\sqrt{xy}}{\sqrt{x}-\sqrt{y}}\div\frac{1}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{x-2\sqrt{xy}+y}{\sqrt{x}-\sqrt{y}}\div\frac{1}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{\left(\sqrt{x}-\sqrt{y}\right)^2}{\sqrt{x}-\sqrt{y}}\times\left(\sqrt{x}-\sqrt{y}\right)\)
\(=x-y\)
\(Q=\frac{x+y-2\sqrt{xy}}{\sqrt{x}-\sqrt{y}}\div\frac{1}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{x-2\sqrt{xy}+y}{\sqrt{x}-\sqrt{y}}\div\frac{1}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{\left(\sqrt{x}-\sqrt{y}\right)^2}{\sqrt{x}-\sqrt{y}}\times\left(\sqrt{x}-\sqrt{y}\right)\)
\(=x-y\)
\(\sqrt{x+2}+x=4\)
\(\text{Đ}K\text{X}\text{Đ}\)\(:\) \(x\ge-2\)
\(\sqrt{x+2}=4-x\)
\(\Leftrightarrow\hept{\begin{cases}4-x\ge0\\x+2=\left(4-x\right)^2\end{cases}}\)
\(\Leftrightarrow x=2\)
................
ĐK: \(-2\le x\le4\)
\(\sqrt{x+2}+x=4\)
\(\Leftrightarrow\)\(\sqrt{x+2}=4-x\)
\(\Leftrightarrow\)\(x+2=16-8x+x^2\)
\(\Leftrightarrow\)\(x^2-9x+14=0\)
\(\Leftrightarrow\)\(\left(x-2\right)\left(x-7\right)=0\)
\(\Leftrightarrow\)\(\orbr{\begin{cases}x=2\left(t/m\right)\\x=7\left(L\right)\end{cases}}\)
Vậy....
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