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Áp dụng bất đẳng thức Cauchy-Schwarz ta có:
\(\sqrt{\left(a+b\right)\left(a+c\right)}\ge\sqrt{a}.\sqrt{a}+\sqrt{b}.\sqrt{c}\)
\(\Leftrightarrow\sqrt{\left(a+b\right)\left(a+c\right)}\ge a+\sqrt{bc}\)
Do đó \(\sqrt{\frac{bc}{\left(c+a\right)\left(a+b\right)}}=\frac{\sqrt{bc\left(c+a\right)\left(a+b\right)}}{\left(c+a\right)\left(a+b\right)}\ge\sqrt{abc}\frac{\sqrt{a}}{\left(c+a\right)\left(c+b\right)}+\frac{bc}{\left(c+a\right)\left(c+b\right)}\left(1\right)\)
Chứng minh tương tự ta được:
\(\hept{\begin{cases}\sqrt{\frac{bc}{\left(c+b\right)\left(a+b\right)}}=\frac{\sqrt{bc\left(c+b\right)\left(a+b\right)}}{\left(c+b\right)\left(a+b\right)}\ge\sqrt{abc}\frac{\sqrt{b}}{\left(c+b\right)\left(a+b\right)}+\frac{ac}{\left(c+b\right)\left(a+b\right)}\left(2\right)\\\sqrt{\frac{ca}{\left(c+a\right)\left(a+b\right)}}=\frac{\sqrt{ca\left(c+a\right)\left(a+b\right)}}{\left(c+a\right)\left(a+b\right)}\ge\sqrt{abc}\frac{\sqrt{c}}{\left(c+a\right)\left(a+b\right)}+\frac{ab}{\left(a+c\right)\left(a+b\right)}\left(3\right)\end{cases}}\)
\(\Rightarrow\sqrt{\frac{bc}{\left(c+a\right)\left(a+b\right)}}+\sqrt{\frac{ca}{\left(c+b\right)\left(a+b\right)}}+\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}\ge\)
\(\sqrt{abc}\left(\frac{\sqrt{a}}{\left(a+c\right)\left(a+b\right)}+\frac{\sqrt{b}}{\left(c+b\right)\left(a+b\right)}+\frac{\sqrt{c}}{\left(c+b\right)\left(a+c\right)}\right)+\)\(\frac{bc}{\left(a+c\right)\left(a+b\right)}+\frac{ac}{\left(c+b\right)\left(a+b\right)}+\frac{ab}{\left(c+b\right)\left(a+c\right)}\left(4\right)\)
Ta lại có: \(\frac{bc}{\left(a+c\right)\left(a+b\right)}+\frac{ac}{\left(c+b\right)\left(a+b\right)}+\frac{ab}{\left(c+b\right)\left(a+c\right)}+\frac{2abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
\(=\frac{bc\left(b+c\right)+ac\left(a+c\right)+ab\left(a+b\right)+2abc}{\left(a+c\right)\left(b+c\right)\left(a+b\right)}\)
\(=\frac{bc\left(a+b+c\right)+ca\left(a+b+c\right)+ab\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\frac{c\left(a+b+c\right)\left(b+a\right)+ab\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
\(=\frac{\left(a+b\right)\left[c\left(a+c\right)+b\left(a+c\right)\right]}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\frac{\left(a+b\right)\left(c+b\right)\left(a+c\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=1\)
\(\left(4\right)\Leftrightarrow\sqrt{\frac{bc}{\left(c+a\right)\left(a+b\right)}}+\sqrt{\frac{ca}{\left(c+b\right)\left(a+b\right)}}+\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}\)\(\ge\sqrt{abc}\left(\frac{\sqrt{a}}{\left(c+a\right)\left(a+b\right)}+\frac{\sqrt{b}}{\left(c+b\right)\left(a+b\right)}+\frac{\sqrt{c}}{\left(c+b\right)\left(a+c\right)}\right)+1-\frac{2abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
Do đó ta cần chứng minh \(\sqrt{abc}\left(\frac{\sqrt{a}}{\left(c+a\right)\left(a+b\right)}+\frac{\sqrt{b}}{\left(c+b\right)\left(a+b\right)}+\frac{\sqrt{c}}{\left(c+b\right)\left(a+c\right)}\right)+1-\frac{2abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)\(\ge1+\frac{4abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
Điều này tương đương với \(\sqrt{a}\left(b+c\right)+\sqrt{b}\left(a+c\right)+\sqrt{c}\left(a+b\right)\ge6\sqrt{abc}\left(5\right)\)
Theo bất đẳng thức AM-GM thì (5) luôn đúng
Dấu "=" xảy ra khi (1);(2);(3) và (5) xảy ra dấu "=". điều này tương đương với a=b=c
Vậy ta có điều phải chứng minh
=))
đk: \(x\ge0;y-z\ge0;z-x\ge0\Leftrightarrow y\ge z\ge x\ge0\)
Ta có: \(pt\Leftrightarrow2\sqrt{x}+2\sqrt{y-z}+2\sqrt{z-x}=x+y-z+z-x+3\)
\(\Leftrightarrow\left(\sqrt{x}-1\right)^2+\left(\sqrt{y-z}-1\right)^2+\left(\sqrt{z-x}-1\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}\sqrt{x}=1\\\sqrt{y-z}=1\\\sqrt{z-x}=1\end{cases}\Leftrightarrow\hept{\begin{cases}x=1\\y=3\\z=2\end{cases}\left(tm\text{đ}k\right)}}\)
\(P=\frac{1}{\sqrt{x}+1}+\frac{10}{2\sqrt{x}+1}-\frac{5}{2x+3\sqrt{x}+1}\)
\(=\frac{1}{\sqrt{x}+1}+\frac{10}{2\sqrt{x}+1}-\frac{5}{\left(2\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{2\sqrt{x}+1+10\left(\sqrt{x}+1\right)-5}{\left(2\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{2\sqrt{x}+1+10\sqrt{x}+10-5}{\left(2\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{6}{\sqrt{x}+1}\)
b) Để P nguyên tố thì \(\frac{6}{\sqrt{x}+1}\) nguyên tố
Để \(P\inℕ^∗\) thì \(\sqrt{x}+1\inƯ\left(6\right)\)
Mà P nguyên tố \(\Rightarrow\frac{6}{\sqrt{x}+1}=\left\{2;3\right\}\Rightarrow\sqrt{x}+1=\left\{2;3\right\}\)
Với \(\sqrt{x}+1=2\Leftrightarrow\sqrt{x}=1\Leftrightarrow x=1\)
Với \(\sqrt{x}+1=3\Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\)
Vậy ...........
