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\(x+y+z+35=2\left(2\sqrt{x+1}+3\sqrt{y+2}+4\sqrt{z+3}\right)\)
\(\Leftrightarrow x+y+z+35-4\sqrt{x+1}-6\sqrt{y+2}-8\sqrt{z+3}=0\)
\(\Leftrightarrow\left(x+1-4\sqrt{x+1}+4\right)+\left(y+2-6\sqrt{y+2}+9\right)+\left(z+3-8\sqrt{z+3}+16\right)=0\)
\(\Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}\left(\sqrt{x+1}-2\right)^2=0\\\left(\sqrt{y+2}-3\right)^2=0\\\left(\sqrt{z+3}-4\right)^2=0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}\sqrt{x+1}=2\\\sqrt{y+2}=3\\\sqrt{z+3}=4\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x=3\\y=7\\z=13\end{cases}}\)
ĐKXĐ : \(\left\{{}\begin{matrix}x+1\ge0\\y+2\ge0\\z+3\ge0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x\ge-1\\y\ge-2\\z\ge-3\end{matrix}\right.\)
Ta có : \(x+y+z+35=2\left(2\sqrt{x+1}+3\sqrt{y+2}+4\sqrt{z+3}\right)\)
=> \(x+y+z+35=4\sqrt{x+1}+6\sqrt{y+2}+8\sqrt{z+3}\)
=> \(x-4\sqrt{x+1}+y-6\sqrt{y+2}+z-8\sqrt{z+3}+35=0\)
=> \(x+1-2.2\sqrt{x+1}+4+y+2-2.3\sqrt{y+2}+9+z+3-4.2\sqrt{z+3}+16=0\)
=> \(\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
Ta thấy : \(\left\{{}\begin{matrix}\left(\sqrt{x+1}-2\right)^2\ge0\\\left(\sqrt{y+2}-3\right)^2\ge0\\\left(\sqrt{z+3}-4\right)^2\ge0\end{matrix}\right.\)
=> \(\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2\ge0\)
- Dấu "=" xảy ra
<=> \(\left\{{}\begin{matrix}\sqrt{x+1}-2=0\\\sqrt{y+2}-3=0\\\sqrt{z+3}-4=0\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}\sqrt{x+1}=2\\\sqrt{y+2}=3\\\sqrt{z+3}=4\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}x+1=4\\y+2=9\\z+3=16\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}x=3\\y=7\\z=13\end{matrix}\right.\) ( TM )
Vậy ...
\(ĐKXĐ:\left\{{}\begin{matrix}x\ge-1\\y\ge-2\\z\ge-3\end{matrix}\right.\)
\(PT\Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x+1}=2\\\sqrt{y+2}=3\\\sqrt{z+3}=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=7\\z=13\end{matrix}\right.\)
ĐKXĐ: ....
\(x+1-4\sqrt{x+1}+4+y+2-6\sqrt{y+2}+9+z+3-8\sqrt{z+3}+16=0\)
\(\Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x+1}=2\\\sqrt{y+2}=3\\\sqrt{z+3}=4\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=3\\y=7\\z=13\end{matrix}\right.\)
ĐKXĐ: \(\left\{{}\begin{matrix}x\ge-1\\y\ge-2\\z\ge-3\end{matrix}\right.\)
x+y+z+35=\(4\sqrt{x+1}+6\sqrt{y+2}+8\sqrt{z+3}\)
\(\Leftrightarrow\left(x+1-2.2\sqrt{x+1}+4\right)+\left(y+2-2.3\sqrt{y+2}+9\right)+\left(z+3-2.4\sqrt{z+3}+16\right)=0\)
\(\Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(\sqrt{x+1}-2\right)^2=0\\\left(\sqrt{y+2}-3^{ }\right)^2=0\\\left(\sqrt{z+3}-4\right)^2=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=7\\z=13\end{matrix}\right.\)
a) DK: x>=2; y>=3; z>=5
\(\Leftrightarrow\left(x-2-2\sqrt{x-2}+1\right)+\left(y-3-2\sqrt{y-3}\cdot2+4\right)+\left(z-5-2\sqrt{z-5}\cdot3+9\right)=0\)
\(\Leftrightarrow\left(\sqrt{x-2}-1\right)^2+\left(\sqrt{y-3}-2\right)^2+\left(\sqrt{z-5}-3\right)^2=0\)(*)
VT(*) >= 0 với mọi x;y;z TMĐK nên để thỏa mãn (*) thì:
\(\hept{\begin{cases}\sqrt{x-2}=1\\\sqrt{y-3}=2\\\sqrt{z-5}=3\end{cases}\Leftrightarrow\hept{\begin{cases}x=3\\y=7\\z=14\end{cases}}}\)
b) x;y;z là nghiệm của PT: \(\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{x+y+z}{2}\left(1\right)\) (1)=> đk: x >=0; y >= 1 ; z >= 2.
Ta có:
- \(\left(\sqrt{x}-1\right)^2\ge0\Rightarrow x-2\sqrt{x}+1\ge0\Rightarrow\sqrt{x}\le\frac{x+1}{2}\)(a)
- Tương tự: \(\sqrt{y-1}\le\frac{y-1+1}{2}=\frac{y}{2}\) (b)
- và: \(\sqrt{z-2}\le\frac{z-2+1}{2}=\frac{z-1}{2}\) (c)
- Do đó: \(\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{x+1+y+z-1}{2}=\frac{x+y+z}{2}\)hay VT(1) <= VP(1) với mọi x;y;z.
Vậy để (1) thỏa mãn thì dấu "=" xảy ra hay các BĐT (a); (b); (c) xảy ra. Khi đó, x = 1; y = 2; z = 3
chả cần HĐT dùng Cosi cx đc
\(\left(x+1\right)+4\ge4\sqrt{x+1}\)
\(\left(y+2\right)+9\ge6\sqrt{y+2}\)
\(\left(z+3\right)+16\ge8\sqrt{z+3}\)
\(\Rightarrow VT\ge VP\).Dấu = khi x=3;y=7;z=13
\(\Leftrightarrow x+y+z+35=4\sqrt{x+1}+6\sqrt{y+2}+8\sqrt{z+3}\)
\(\Leftrightarrow x+1-4\sqrt{x+1}+4+y+2-6\sqrt{y+2}+9+z+3-8\sqrt{z+3}+16=0\)
\(\Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
cho từng cái ngoặc bằng 0 thì ta được x=3 ; y=7 ;z=13 nếu đúng tick nha bạn