\(P=\dfrac{x\left(\sqrt{y}-\sqrt{z}\right)-y\left(\sqrt{x}-\sqrt{z}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}+\dfrac{z}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

\(=\dfrac{x\sqrt{y}-x\sqrt{z}-y\sqrt{x}+y\sqrt{z}+z\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

\(=\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)-\sqrt{z}\left(x-y\right)+z\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

\(=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{xy}-\sqrt{z}\left(\sqrt{x}+\sqrt{y}\right)+z\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

\(=\dfrac{\left(\sqrt{xy}-\sqrt{zx}-\sqrt{zy}+z\right)}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{y}-\sqrt{z}\right)-\sqrt{z}\left(\sqrt{y}-\sqrt{z}\right)}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

\(=\dfrac{\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{x}-\sqrt{z}\right)}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\)

=1

phân số thứ 3 sai

Đặt \(\left\{{}\begin{matrix}x-4=a\\y-3=b\end{matrix}\right.\) \(\Rightarrow a^2+b^2=5\)

\(Q=\sqrt{\left(a+5\right)^2+b^2}+\sqrt{\left(a+3\right)^2+\left(b+4\right)^2}\)

\(\Rightarrow Q\le\sqrt{2\left[\left(a+5\right)^2+b^2+\left(a+3\right)^2+\left(b+4\right)^2\right]}\) (Bunhiacopxki)

\(\Rightarrow Q\le\sqrt{4\left(a^2+8a+b^2+4b+25\right)}\)

\(\Rightarrow Q\le\sqrt{4\left(a^2+2.4a+b^2+2.2b+25\right)}\)

\(\Rightarrow Q\le\sqrt{4\left(a^2+2\left(a^2+4\right)+b^2+2\left(b^2+1\right)+25\right)}\)

\(\Rightarrow Q\le\sqrt{4\left(3a^2+3b^2+35\right)}\le\sqrt{4\left(3.5+35\right)}=10\sqrt{2}\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}a=2\\b=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=6\\y=4\end{matrix}\right.\)

\(A=\frac{\left(y+z\right)\sqrt{\left(x+y\right)\left(x+z\right)}}{x}+\frac{\left(x+z\right)\sqrt{\left(x+y\right)\left(y+z\right)}}{y}+\frac{\left(x+y\right)\sqrt{\left(y+z\right)\left(x+z\right)}}{z}.\)

Áp dụng bất đẳng thức Bunhiacopski ta có

\(\left(x+y\right)\left(x+z\right)\ge\left(x+\sqrt{yz}\right)^2\)

Tương tự \(\left(x+y\right)\left(y+z\right)\ge\left(y+\sqrt{xz}\right)^2\)

                 \(\left(y+z\right)\left(x+z\right)\ge\left(z+\sqrt{xy}\right)^2\)

\(\Rightarrow A\ge\frac{\left(y+z\right)\left(x+\sqrt{yz}\right)}{x}+\frac{\left(x+z\right)\left(y+\sqrt{xz}\right)}{y}+\frac{\left(x+y\right)\left(z+\sqrt{xy}\right)}{z}\)

hay \(A\ge2\left(x+y+z\right)+\frac{\sqrt{yz}\left(y+z\right)}{x}+\frac{\left(x+z\right)\sqrt{xz}}{y}+\frac{\left(x+y\right)\sqrt{xy}}{z}\)

\(\Leftrightarrow A\ge2\left(x+y+z\right)+\frac{yz\sqrt{yz}\left(y+z\right)}{xyz}+\frac{xz\sqrt{xz}\left(x+z\right)}{xyz}+\frac{xy\sqrt{xy}\left(x+y\right)}{xyz}\)

Đặt \(M=\frac{yz\sqrt{yz}\left(y+z\right)}{xyz}+\frac{xz\sqrt{xz}\left(x+z\right)}{xyz}+\frac{xy\sqrt{xy}\left(x+y\right)}{xyz}\)

Ta có \(\left(x,y,z\right)\rightarrow\left(a^2,b^2,c^2\right)\)

Khi đó \(M=\frac{a^3b^3\left(a^2+b^2\right)+b^3c^3\left(b^2+c^2\right)+c^3a^3\left(a^2+c^2\right)}{a^2b^2c^2}\)

ÁP DỤNG BĐT AM-GM ta có

\(a^5b^3+a^3b^5\ge2\sqrt{a^8b^8}=2a^4b^4\)

\(b^5c^3+b^3c^5\ge2\sqrt{b^8c^8}=2b^4c^4\)

\(a^5c^3+a^3c^5\ge2\sqrt{a^8c^8}=2a^4c^4\)

Cộng từng vế ta được 

\(a^3b^3\left(a^2+b^2\right)+b^3c^3\left(b^2+c^2\right)+c^3a^3\left(a^2+c^2\right)\ge2\left(a^4b^4+b^4c^4+c^4a^4\right)\)

              \(\ge2a^2b^2c^2\left(a^2+b^2+c^2\right)\)

\(\Rightarrow M\ge2\left(a^2+b^2+c^2\right)=2\left(x+y+z\right)\)

\(\Rightarrow A\ge4\left(x+y+z\right)=4\sqrt{2019}\)

Dấu "=" xảy ra khi \(x=y=z=\frac{\sqrt{2019}}{3}\)

mình cần gấp, thanks các bạn

Đề chắc chắn đúng chứ bạn??

đúng đề bạn ạ

a) ĐKXĐ: \(x,y\ge0\)

\(M=\dfrac{x\sqrt{y}-\sqrt{y}-y\sqrt{x}+\sqrt{x}}{1+\sqrt{xy}}=\dfrac{x\sqrt{y}-y\sqrt{x}+\sqrt{x}-\sqrt{y}}{1+\sqrt{xy}}\)

\(=\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)+\left(\sqrt{x}-\sqrt{y}\right)}{1+\sqrt{xy}}=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(1+\sqrt{xy}\right)}{1+\sqrt{xy}}=\sqrt{x}-\sqrt{y}\)

b) \(x=\left(1-\sqrt{3}\right)^2\Rightarrow\sqrt{x}=\sqrt{\left(1-\sqrt{3}\right)^2}=\left|1-\sqrt{3}\right|=\sqrt{3}-1\)

\(y=3-\sqrt{8}\Rightarrow\sqrt{y}=\sqrt{3-\sqrt{8}}=\sqrt{3-2\sqrt{2}}=\sqrt{\left(\sqrt{2}\right)^2-2.\sqrt{2}.1+1^2}\)

\(=\sqrt{\left(\sqrt{2}-1\right)^2}=\left|\sqrt{2}-1\right|=\sqrt{2}-1\)

\(\Rightarrow M=\left(\sqrt{3}-1\right)-\left(\sqrt{2}-1\right)=\sqrt{3}-\sqrt{2}\)