665+1
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ĐK : \(x\ge3;y\ge1;z\ge665\)
\(\dfrac{16}{\sqrt{x-3}}+\dfrac{4}{\sqrt{y-1}}+\dfrac{1225}{\sqrt{z-665}}=82-\sqrt{x-3}-\sqrt{y-1}-\sqrt{z-665}\)
\(\Leftrightarrow\left(\dfrac{16}{\sqrt{x-3}}+\sqrt{x-3}\right)+\left(\dfrac{4}{\sqrt{y-1}}+\sqrt{y-1}\right)+\left(\dfrac{1225}{\sqrt{z-665}}+\sqrt{z-665}\right)=82\)
Theo BĐT Cô Si cho các số dương ta có :
\(\left\{{}\begin{matrix}\dfrac{16}{\sqrt{x-3}}+\sqrt{x-3}\ge2\sqrt{\dfrac{16\sqrt{x-3}}{\sqrt{x-3}}}=2\sqrt{16}=8\\\dfrac{4}{\sqrt{y-1}}+\sqrt{y-1}\ge2\sqrt{\dfrac{4\sqrt{y-1}}{\sqrt{y-1}}}=2\sqrt{4}=4\\\dfrac{1225}{\sqrt{z-665}}+\sqrt{z-665}\ge2\sqrt{\dfrac{1225\sqrt{z-665}}{\sqrt{z-665}}}=2\sqrt{1225}=70\end{matrix}\right.\)
\(\Rightarrow\left(\dfrac{16}{\sqrt{x-3}}+\sqrt{x-3}\right)+\left(\dfrac{4}{\sqrt{y-1}}+\sqrt{y-1}\right)+\left(\dfrac{1225}{\sqrt{z-665}}+\sqrt{z-665}\right)\ge82\)
Dấu \("="\) hiển nhiên xảy ra khi :
\(\left\{{}\begin{matrix}\dfrac{16}{\sqrt{x-3}}=\sqrt{x-3}\\\dfrac{4}{\sqrt{y-1}}=\sqrt{y-1}\\\dfrac{1225}{\sqrt{z-665}}=\sqrt{z-665}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-3=16\\y-1=4\\z-665=1225\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=19\\y=5\\z=1890\end{matrix}\right.\)
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Hướng dẫn:
Biến đổi về dạng: \(\frac{\left(4-\sqrt{x-3}\right)^2}{\sqrt{x-3}}+\frac{\left(2-\sqrt{y-1}\right)^2}{\sqrt{y-1}}+\frac{\left(35-\sqrt{z-665}\right)^2}{\sqrt{z-665}}=0\)
\(\Leftrightarrow\hept{\begin{cases}\sqrt{x-3}=4\\\sqrt{y-1}=2\\\sqrt{z-665}=35\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=9\\y=5\\z=1890\end{cases}}\)
Đây là câu trả lời cho bạn nào cần thiết bài này !
đk: \(\hept{\begin{cases}x\ge3\\y\ge1\\z\ge665\end{cases}}\)
Ta có: \(\frac{16}{\sqrt{x-3}}+\frac{4}{\sqrt{y-1}}+\frac{1225}{\sqrt{z-665}}=82-\sqrt{x-3}-\sqrt{y-1}-\sqrt{x-665}\)
<=> \(\left(\frac{16}{\sqrt{x-3}}+\sqrt{x-3}\right)+\left(\frac{4}{\sqrt{y-1}}+\sqrt{y-1}\right)+\left(\frac{1225}{\sqrt{z-665}}+\sqrt{z-665}\right)=82\)
Mà \(VT\ge2\sqrt{\frac{16}{\sqrt{x-3}}\cdot\sqrt{x-3}}+2\sqrt{\frac{4}{\sqrt{y-1}}\cdot\sqrt{y-1}}+2\sqrt{\frac{1225}{\sqrt{z-665}}\cdot\sqrt{z-665}}\)
\(=2\cdot4+2\cdot2+2\cdot35=82\left(\forall x,y,z\right)\)
Dấu "=" xảy ra khi: \(\frac{16}{\sqrt{x-3}}=\sqrt{x-3}\) ; \(\frac{4}{\sqrt{y-1}}=\sqrt{y-1}\) ; \(\frac{1225}{\sqrt{z-665}}=\sqrt{z-665}\)
GPT ra ta sẽ được: \(\hept{\begin{cases}x=19\\y=5\\z=1890\end{cases}}\)
Vậy \(\left(x;y;z\right)=\left(19;5;1890\right)\) sinh nhật Bác luôn đấy ạ:))
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665+1=666
665 + 1 = 666
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