Tìm x,y sao cho \(\left(x-12+y\right)^{200}+\left(x-4-y\right)^{200}\le0\)
K
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24 tháng 7 2017
\(\left(x-3\right)^2+\left(y+2\right)^2=0\)
\(\left\{{}\begin{matrix}\left(x-3\right)^2\ge0\forall x\\\left(y+2\right)^2\ge0\forall y\end{matrix}\right.\)
\(\Rightarrow\left(x-3\right)^2+\left(y+2\right)^2\ge0\)
Dấu "=" xảy ra khi:
\(\left\{{}\begin{matrix}\left(x-3\right)^2=0\Rightarrow x-3=0\Rightarrow x=3\\\left(y+2\right)^2=0\Rightarrow y+2=0\Rightarrow y=-2\end{matrix}\right.\)
đề sai câu b các câu sau áp dụng tương tự
24 tháng 7 2017
c/ Vì: \(\left(x-12+y\right)^{200}+\left(x-4-x\right)^{200}=0\)
mà \(\left\{{}\begin{matrix}\left(x-12+y\right)^{200}\ge0\forall x,y\\\left(x-4-y\right)^{200}\ge0\forall x,y\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(x-12+y\right)^{200}=0\\\left(x-4-y\right)^{200}=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-12+y=0\\x-4-y=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x+y=12\\x-y=4\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=8\\y=4\end{matrix}\right.\)
G6
1
28 tháng 10 2018
\(\left(3x-5\right)^{100}\ge0;\left(2y+1\right)^{200}\ge0\)
\(\Rightarrow\left(3x-5\right)^{10}+\left(2y+1\right)^{200}\ge0\)
Dấu "=" xảy ra khi \(\hept{\begin{cases}3x-5=0\\2y+1=0\end{cases}\Leftrightarrow}\hept{\begin{cases}x=\frac{5}{3}\\y=-\frac{1}{2}\end{cases}}\)
LT
1
11 tháng 7 2018
\(\hept{\begin{cases}\left(3x-5\right)^{100}\ge0\\\left(2y+3\right)^{200}\ge0\end{cases}}\)\(\Rightarrow\left(3x-5\right)^{100}+\left(2y+3\right)^{200}\ge0\)
Kết hợp với giả thiết:\(\hept{\begin{cases}\left(3x-5\right)^{100}=0\\\left(2y+3\right)^{200}=0\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}3x-5=0\\2y+3=0\end{cases}}\)\(\Rightarrow\hept{\begin{cases}3x=5\\2y=-3\end{cases}}\)\(\Rightarrow\hept{\begin{cases}x=\frac{5}{3}\\y=-\frac{3}{2}\end{cases}}\)
KN
5 tháng 8 2019
a) \(5^{3x+1}=25^{x+2}\)
\(\Leftrightarrow5^{3x+1}=\left(5^2\right)^{x+2}\)
\(\Leftrightarrow5^{3x+1}=5^{2x+4}\)
\(\Leftrightarrow3x+1=2x+4\)
\(\Leftrightarrow3x-2x=4-1\)
\(\Leftrightarrow x=3\)
15 tháng 9 2021
a) \(x^2+\left(y-\dfrac{1}{10}\right)^4=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\y-\dfrac{1}{10}=0\end{matrix}\right.\)( do \(x^2\ge0,\left(y-\dfrac{1}{10}\right)^4\ge0\))
\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=\dfrac{1}{10}\end{matrix}\right.\)
b) \(\left(\dfrac{1}{2}.x-5\right)^{20}+\left(y^2-\dfrac{1}{4}\right)^{10}\le0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{2}x-5=0\\y^2-\dfrac{1}{4}=0\end{matrix}\right.\)( do \(\left(\dfrac{1}{2}x-5\right)^{20}\ge0,\left(y^2-\dfrac{1}{4}\right)^{10}\ge0\))
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{2}x=5\\y^2=\dfrac{1}{4}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=10\\y=\pm\dfrac{1}{2}\end{matrix}\right.\)
15 tháng 9 2021
\(a,\Leftrightarrow\left\{{}\begin{matrix}x=0\\y-\dfrac{1}{10}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=\dfrac{1}{10}\end{matrix}\right.\\ b,\left\{{}\begin{matrix}\left(\dfrac{1}{2}x-5\right)^{20}\ge0\\\left(y^2-\dfrac{1}{4}\right)^{10}\ge0\end{matrix}\right.\Leftrightarrow\left(\dfrac{1}{2}x-5\right)^{20}+\left(y^2-\dfrac{1}{4}\right)^{10}\ge0\)
Mà \(\left(\dfrac{1}{2}x-5\right)^{20}+\left(y^2-\dfrac{1}{4}\right)^{10}\le0\)
\(\Leftrightarrow\left(\dfrac{1}{2}x-5\right)^{20}+\left(y^2-\dfrac{1}{4}\right)^{10}=0\\ \Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{2}x=5\\y^2=\dfrac{1}{4}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=10\\y=\pm\dfrac{1}{2}\end{matrix}\right.\)
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
18 tháng 7 2023
(\(x-3\))2 + (2y - 1)2 = 0
(\(x\) - 3)2 ≥ 0 ∀ \(x\)
(2y - 1)2 ≥ 0 ∀ y
⇔ (\(x\) - 3)2 + (2y - 1)2= 0
⇔ \(\left\{{}\begin{matrix}x-3=0\\3y-1=0\end{matrix}\right.\)
\(\Leftrightarrow\) \(\left\{{}\begin{matrix}x=2\\y=\dfrac{1}{3}\end{matrix}\right.\)
(4\(x-3\))4 + (y + 2)2 ≤ 0
(4\(x\) - 3)4 ≥ 0 ∀ \(x\)
(y + 2)2 ≥ 0 ∀ y
⇔(4\(x\) - 3)4 + (y+2)2 ≥ 0
⇔ (4\(x\) - 3)4 + (y + 2)2 ≤ 0 ⇔
⇔\(\left\{{}\begin{matrix}4x-3=0\\y+2=0\end{matrix}\right.\)
⇔ \(\left\{{}\begin{matrix}x=\dfrac{3}{4}\\y=-2\end{matrix}\right.\)
25 tháng 7 2020
\(\frac{1}{3}.3^n+5.3^{n-1}=162\)
<=> \(3^{n-1}+5.3^{n-1}=162\)
<=> \(3^{n-1}\left(1+5\right)=162\)
<=> \(3^{n-1}.6=162\)
<=> \(3^{n-1}=162:6\)
<=> \(3^{n-1}=27\)
<=> \(3^{n-1}=3^3\)
<=> n - 1 = 3
<=> n = 3 + 1 = 4
Y
25 tháng 7 2020
Câu 1
a) Từ gt=>\(\hept{\begin{cases}x-5=1-3x\\x-5=3x-1\end{cases}}\)
<=>\(\hept{\begin{cases}4x=6\\2x=-4\end{cases}}\)
<=>\(\hept{\begin{cases}x=\frac{3}{2}\\x=-2\end{cases}}\)
b) Ta có: \(\hept{\begin{cases}\left(3x-1\right)^{100}\ge0,\forall x\in R\\\left(2y+1\right)^{200}\ge0,\forall x\in R\end{cases}}\)
Kết hợp với đề bài => \(\hept{\begin{cases}3x-1=0\\2y+1=0\end{cases}}\)
=>\(\hept{\begin{cases}x=\frac{1}{3}\\y=-\frac{1}{2}\end{cases}}\)
Bài 2
\(\frac{1}{3}.3^n+5.3^{n-1}=162\)
<=>\(3^{n-1}+5.3^{n-1}=162\)
<=>\(6.3^{n-1}=162\)
<=>\(3^{n-1}=27=3^3\)
<=>\(n-1=3\)
<=>\(n=4\)
28 tháng 9 2021
a) \(\left|3x-\dfrac{1}{2}\right|+\left|\dfrac{1}{4}y+\dfrac{3}{5}\right|=0\)
Do \(\left|3x-\dfrac{1}{2}\right|,\left|\dfrac{1}{4}y+\dfrac{3}{5}\right|\ge0\forall x,y\)
\(\Rightarrow\left\{{}\begin{matrix}3x-\dfrac{1}{2}=0\\\dfrac{1}{4}y+\dfrac{3}{5}=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{1}{6}\\y=-\dfrac{12}{5}\end{matrix}\right.\)
b) \(\left|\dfrac{3}{2}x+\dfrac{1}{9}\right|+\left|\dfrac{5}{7}y-\dfrac{1}{2}\right|\le0\)
Do \(\left|\dfrac{3}{2}x+\dfrac{1}{9}\right|,\left|\dfrac{5}{7}y-\dfrac{1}{2}\right|\ge0\forall x,y\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{3}{2}x+\dfrac{1}{9}=0\\\dfrac{5}{7}y-\dfrac{1}{2}=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=-\dfrac{2}{27}\\y=\dfrac{7}{10}\end{matrix}\right.\)
Vì: \(\left(x-12+y\right)^{200}\ge0;\left(x-4-y\right)^{200}\ge0\)
=> \(\left(x-12+y\right)^{200}+\left(x-4-y\right)^{200}\le0\)
\(\Leftrightarrow\left(x-12+y\right)^{200}+\left(x-4-y\right)^{200}=0\)
\(\Leftrightarrow\begin{cases}x-12+y=0\\x-4-y=0\end{cases}\)\(\Leftrightarrow\begin{cases}x+y=12\\x-y=4\end{cases}\)\(\Leftrightarrow\begin{cases}x=8\\y=4\end{cases}\)