Tìm GTNN của biểu thức
\(M=\sqrt{x^2+y^2-2xy+2x-2y+10}+2y^2-8y+2024\)
\(Q=\sqrt{25x^2-20x+4}+\sqrt{25x^2-30x+9}\)
K
Khách
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Những câu hỏi liên quan
28 tháng 8 2022
\(B=\left|5x-2\right|+\left|5x-3\right|\)
\(=\left|5x-2\right|+\left|3-5x\right|\)
=>B>=|5x-2+3-5x|=1
Dấu = xảy ra khi (5x-2)(5x-3)<=0
=>2/5<=x<=3/5
NV
Nguyễn Việt Lâm
Giáo viên
19 tháng 10 2019
1/
\(\Leftrightarrow\left\{{}\begin{matrix}2x-y=0\\y-2=0\\x+y+z=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=1\\y=2\\z=-3\end{matrix}\right.\)
2/ \(P=\sqrt{\left(5x-2\right)^2}+\sqrt{\left(3-5x\right)^2}\)
\(P=\left|5x-2\right|+\left|3-5x\right|\ge\left|5x-2+3-5x\right|=1\)
\(\Rightarrow P_{min}=1\) khi \(\frac{2}{5}\le x\le\frac{3}{5}\)
3/ ĐKXĐ: \(\left|x\right|\ge1\)
\(x^2-1-\sqrt{x^2-1}=0\)
\(\Leftrightarrow\sqrt{x^2-1}\left(\sqrt{x^2-1}-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2-1}=0\\\sqrt{x^2-1}=1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x^2-1=0\\x^2-1=1\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\pm1\\x=\pm\sqrt{2}\end{matrix}\right.\)
PD
4
MT
2 tháng 2 2016
câu a) rút x theo y thế vào A rồi áp dụng HĐT
b)rút xy thế vào B
c)HĐT
d)rút x theo y thé vào C
rồi dùng BĐT cô-si
e)BĐT chưa dấu giá trị tuyệt đối
LD
2
17 tháng 7 2017
\(=\sqrt{\left(5x-2\right)^2}+\sqrt{\left(5x\right)^2}\)= \(\left|2-5x\right|+\left|5x\right|\ge2+5x-5x=2\)
min A=2 \(\Leftrightarrow\hept{\begin{cases}2-5x\ge0\\5x\ge0\end{cases}\Leftrightarrow0\le x\le\frac{2}{5}}\)
AV
18 tháng 6 2018
Ta có : M =\(\sqrt{\left(5x\right)^2-2.2.5x+2^2}+\sqrt{\left(5x\right)^2}\)= \(\sqrt{\left(5x-2\right)^2}+\left|5x\right|\) = \(\left|5x-2\right|+\left|5x\right|=\left|2-5x\right|+\left|5x\right|\ge\left|2-5x+5x\right|=\left|2\right|=2\)
=> M ≥ 2
Min M = 2 : dấu"="xảy ra khi: ....
Mình bận rồi tự làm tí nhé!!
9 tháng 7 2019
\(\sqrt{x^2+y^2-2xy+2x-2y+5}+2y^2-8y+2015\)
\(=\sqrt{\left(x^2+y^2-2xy\right)+2\left(x-y\right)+1+4}+2\left(y^2-4y+4\right)+2007\)\(=\sqrt{\left(x-y+1\right)^2+4}+2\left(y-2\right)^2+2007\ge2007\)
Dấu "=" xảy ra khi: \(\left\{{}\begin{matrix}x-y+1=0\\y-2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)
\(M=\sqrt{x^2+y^2-2xy+2x-2y+10}+2y^2-8y+2024\\ =\sqrt{\left(x^2+y^2+1-2xy+2x-2y\right)+9}+\left(2y^2-8y+8\right)+2016\\ =\sqrt{\left(x-y+1\right)^2+9}+2\left(y^2-4y+4\right)+2016\\ =\sqrt{\left(x-y+1\right)^2+9}+2\left(y-2\right)^2+2016\) \(\text{Do }\left(x-y+1\right)^2\ge0\forall x;y\\ \Rightarrow\left(x-y+1\right)^2+9\ge9\forall x;y\\ \Rightarrow\sqrt{\left(x-y+1\right)^2+9}\ge3\forall x;y\\ Mà\text{ }2\left(y-2\right)^2\ge0\forall y\\ \Rightarrow\sqrt{\left(x-y+1\right)^2+9}+2\left(y-2\right)^2\ge3\forall x;y\\ M=\sqrt{\left(x-y+1\right)^2+9}+2\left(y-2\right)^2+2016\ge2019\forall x;y\)
Dấu "=" xảy ra khi:
\(\left\{{}\begin{matrix}2\left(y-2\right)^2=0\\\left(x-y+1\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y-2=0\\x-y+1=0\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y=2\\x=y-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=2\\x=1\end{matrix}\right.\)
Vậy \(M_{Min}=2019\) khi \(\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)
\(Q=\sqrt{25x^2-20x+4}+\sqrt{25x^2-30x+9}\\ =\sqrt{\left(5x-2\right)^2}+\sqrt{\left(5x-3\right)^2}\\ =\left|5x-2\right|+\left|5x-3\right|\\ =\left|5x-2\right|+\left|3-5x\right|\)
Áp dụng BDT: \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\)
\(\Rightarrow\left|5x-2\right|+\left|3-5x\right|\ge\left|5x-2+3-5x\right|=\left|1\right|=1\)
Dấu "=" xảy ra khi:
\(\left(5x-2\right)\left(3-5x\right)\ge0\\\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}5x-2\ge0\\3-5x\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}5x-2\le0\\3-5x\le0\end{matrix}\right.\end{matrix}\right. \) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}5x\ge2\\5x\le3\end{matrix}\right.\\\left\{{}\begin{matrix}5x\le2\\5x\ge3\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge\dfrac{2}{5}\\x\le\dfrac{3}{5}\end{matrix}\right.\left(T/m\right)\\\left\{{}\begin{matrix}x\le\dfrac{2}{5}\\x\ge\dfrac{3}{5}\end{matrix}\right.\left(K^0\text{ }T/m\right)\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{2}{5}\le x\le\dfrac{3}{5}\)
Vậy \(Q_{Min}=1\) khi \(\dfrac{2}{5}\le x\le\dfrac{3}{5}\)