Tìm GTNN \(P=\sqrt{x^2-2x+1}+\sqrt{x^2-6x+9}\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
A=\(\sqrt{x^2-2x+1}+\sqrt{x^2+6x+9}=\sqrt{\left(x-1\right)^2}+\sqrt{\left(x+3\right)^2}\)=|x-1|+|x+3|=|1-x|+|x+3|
Áp dụng bđt |a|+|b|\(\ge\)|a+b| ta được: A=|1-x|+|x+3|\(\ge\)|1-x+x+3|=4
Dấu "=" xảy ra khi (1-x)(x+3)\(\ge\)0 <=> \(-3\le x\le1\)
Vậy Amin=4 khi \(-3\le x\le1\)
A = \(\sqrt{x^2-2x+1}+\sqrt{x^2+6x+9}\)
= \(\sqrt{\left(1-x\right)^2}+\sqrt{\left(x+3\right)^2}\)
= 1 - x + x + 3
= 4
![](https://rs.olm.vn/images/avt/0.png?1311)
c/ \(C=\sqrt{x^2-6x+9}+\sqrt{x^2+10x+25}\)
\(=\sqrt{\left(x-3\right)^2}+\sqrt{\left(x+5\right)^2}\)
\(=|3-x|+|x+5|\ge|3-x+x+5|=8\)
d/ \(D=\sqrt{x^2-6x+9}+\sqrt{4x^2+24x+36}\)
\(=\sqrt{\left(x-3\right)^2}+\sqrt{4\left(x+3\right)^2}\)
\(=|3-x|+|x+3|+|x+3|\ge|3-x+x+3|+0=6\)
e/ \(2E=\sqrt{x^2}+2\sqrt{x^2-2x+1}\)
\(=\sqrt{x^2}+2\sqrt{\left(x-1\right)^2}\)
\(=|x|+|1-x|+|x-1|\ge|x+1-x|+0=1\)
\(\Rightarrow E\ge\frac{1}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Tìm đc mỗi GTNN, cách tìm GTLN chưa chắc chắn lắm nên mk ko lm nha :D
1/ \(A=\sqrt{\left(x-1\right)^2}+\sqrt{\left(3-x\right)^2}=\left|x-1\right|+\left|3-x\right|\ge\left|x-1+3-x\right|=2\)
2/ \(B=\sqrt{x-1-2\sqrt{x-1}+1}+\sqrt{x-1+2\sqrt{x-1}+1}=\sqrt{\left(1-\sqrt{x-1}\right)^2}+\sqrt{\left(\sqrt{x-1}+1\right)^2}\)
\(=\left|1-\sqrt{x-1}\right|+\left|\sqrt{x-1}+1\right|\ge\left|1-\sqrt{x-1}+\sqrt{x-1}+1\right|=2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:
\(C=\sqrt{-x^2+6x}\)
Mà: \(\sqrt{-x^2+6x}\ge0\)
Dấu "=" xảy ra khi:
\(\sqrt{-x^2+6x}=0\)
\(\Leftrightarrow\sqrt{-x\left(x-6\right)}=0\)
\(\Leftrightarrow-x\left(x-6\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=6\end{matrix}\right.\)
Vậy: \(C_{min}=0\) khi \(\left[{}\begin{matrix}x=0\\x=6\end{matrix}\right.\)
\(D=\sqrt{6x-2x^2}\)
Mà: \(\sqrt{6x-2x^2}\ge0\)
Dấu "=" xảy ra khi:
\(\sqrt{6x-2x^2}=0\)
\(\Leftrightarrow\sqrt{2x\left(3-x\right)}=0\)
\(\Leftrightarrow2x\left(3-x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)
Vậy: \(D_{min}=0\) khi \(\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Lời giải:
Ta có:
$P=\sqrt{x^2+2x+1}+2\sqrt{x^2-6x+9}$
$=\sqrt{(x+1)^2}+2\sqrt{(x-3)^2}$
$=|x+1|+2|x-3|$
Áp dụng BĐT dạng $|a|+|b|\geq |a+b|$ ta có:
$|x+1|+|x-3|=|x+1|+|3-x|\geq |x+1+3-x|=4$
$|x-3|\geq 0$ theo tính chất trị tuyệt đối
$\Rightarrow P\geq 4$
Vậy GTNN của $P$ là $4$
Dấu "=" xảy ra khi \(\left\{\begin{matrix} (x+1)(3-x)\geq 0\\ x-3=0\end{matrix}\right.\Leftrightarrow x=3\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ngại làm lần 2 quá bạn ơi
Câu hỏi của Chuột yêu Gạo - Toán lớp 9 | Học trực tuyến
![](https://rs.olm.vn/images/avt/0.png?1311)
a) \(A=\sqrt{x^2-2x+1}+\sqrt{x^2-6x+9}\)
\(=\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-3\right)^2}\)
\(=\left|x-1\right|+\left|x-3\right|\ge\left|\left(x-1\right)+\left(3-x\right)\right|=2\)
Vậy\(A_{min}=2\Leftrightarrow\left(x-1\right)\left(3-x\right)\ge0\)
\(TH1:\hept{\begin{cases}x-1\ge0\\3-x\ge0\end{cases}}\Leftrightarrow1\le x\le3\)
\(TH1:\hept{\begin{cases}x-1\le0\\3-x\le0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\le1\\x\ge3\end{cases}}\left(L\right)\)
Vậy \(A_{min}=2\Leftrightarrow1\le x\le3\)
\(P=\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-3\right)^2}=\left|x-1\right|+\left|3-x\right|\ge\left|x-1+3-x\right|=2\)
Dấu "=" xảy ra khi \(\left(x-1\right)\left(3-x\right)\ge0\Leftrightarrow1\le x\le3\)
Vậy GTNN cùa P là 2.