Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Bài 1:
1.Đặt \(A=x^2+y^2-3x+2y+3\)
\(=x^2-2.x.\frac{3}{2}+\frac{9}{4}-\frac{9}{4}+y^2+2y+1+2\)
\(=\left(x-\frac{3}{2}\right)^2+\left(y+1\right)^2-\frac{9}{4}+2\)
\(=\left(x-\frac{3}{2}\right)^2+\left(y+1\right)^2-\frac{1}{4}\)
Vì \(\hept{\begin{cases}\left(x-\frac{3}{2}\right)^2\ge0;\forall x\\\left(y+1\right)^2\ge0;\forall y\end{cases}}\)
\(\Rightarrow\left(x-\frac{3}{2}\right)^2+\left(y+1\right)^2\ge0;\forall x,y\)
\(\Rightarrow\left(x-\frac{3}{2}\right)^2+\left(y+1\right)^2-\frac{1}{4}\ge0-\frac{1}{4};\forall x,y\)
Hay \(A\ge\frac{-1}{4};\forall x,y\)
Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(x-\frac{3}{2}\right)^2=0\\\left(y+1\right)^2=0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x=\frac{3}{2}\\y=-1\end{cases}}\)
VẬY MIN A=\(\frac{-1}{4}\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{3}{2}\\y=-1\end{cases}}\)
\(N = 5x^2 + 2y^ 2 + 4xy - 2x + 4y + 2015\)
\(N = ( 4x^ 2 + 4xy + y ^ 2 ) + ( x^2 - 2x + 1 )+\)
\(( y^2 + 4y + 4 ) + 2010\)
\(N = ( 2x + y )^2 + ( x - 1 )^2 + ( y + 2 )^2 + 2010\)
\(\ge\)\(2010\)
\(Dấu " = " xảy ra \)\(\Leftrightarrow\) \(2x + y = 0 và\)\(x - 1 = 0 và y + 2 = 0\)
\(\Rightarrow\)\(x = 1 và y = - 2\)
\(Min N = 2010\)\(\Leftrightarrow\)\(x = 1 và y = - 2\)
\(\frac{x}{x-2y}+\frac{x}{x+2y}+\frac{4xy}{4y^2-x^2}\)
\(=\frac{x\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}+\frac{x\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}+\frac{-4xy}{\left(x-2y\right)\left(x+2y\right)}\)
\(=\frac{x^2+2xy+x^2-2xy-4xy}{\left(x-2y\right)\left(x+2y\right)}\)
\(=\frac{2x^2-4xy}{\left(x-2y\right)\left(x+2y\right)}\)
TA có :
\(H=x^2+2xy+y^2-2x-2y=\left(x^2+y^2+1+2xy-2x-2y\right)-1=\left(x+y-1\right)^2-1\)
Vì \(\left(x+y-1\right)^2\ge0\) nên \(\left(x+y-1\right)^2-1\ge-1\)
Vậy GTNN của H là -1 khi x+y-1=0 => x+y = 1
BẢO HÙNG HÓM HỈNH LỚP TAO LÀM CHO CÒN TAO CHO Ý H
H=\(X^2+2XY+Y^2-2X-2Y\)
H=\(\left(X+Y\right)^2-2\left(X+Y\right)\)
H=\(\left(X+Y\right)^2\)\(-2.\left(X+Y\right).1+1\))-1
H=\(\left(X+Y-1\right)^2-1\)
VẬY GTNN LÀ -1
a, \(A_{\left(x\right)}=2x^2+2xy+y^2-2x+2y+2\)
\(=\left(x^2+y^2+1+2xy+2x+2y\right)+\left(x^2-4x+4\right)-3\)
\(=\left(x+y+1\right)^2+\left(x-2\right)^2-3\ge-3\) hay \(A_{\left(x\right)}\ge-3\)
Dấu ''='' xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}\left(x+y+1\right)^2=0\\\left(x-2\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x+y+1=0\\x-2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}y=-3\\x=2\end{matrix}\right.\)
Vậy \(minA_{\left(x\right)}=-3\) khi x=-3; y=2
b, \(B_{\left(x\right)}=x^2-4xy+5y^2+10x-22y+28\)
\(=\left(x^2+4y^2+25-4xy+10x-20y\right)+\left(y^2-2y+1\right)+2\)
\(=\left(x-2y+5\right)^2+\left(y-1\right)^2+2\ge2\Leftrightarrow B_{\left(x\right)}\ge2\)
Dấu ''='' xảy ra khi \(\left\{{}\begin{matrix}\left(x-2y+5\right)^2=0\\\left(y-1\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-2y+5=0\\y-1=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)
Vậy \(minB_{\left(x\right)}=2\Leftrightarrow x=-3;y=1\)
c, \(C_{\left(x\right)}=x^2-10xy+26y^2+14x-76y+59\)
\(=\left(x^2+25y^2+49-10xy+14x-70y\right)+\left(y^2-6y+9\right)+1\)
\(=\left(x-5y+7\right)^2+\left(y-3\right)^2+1\ge1\Leftrightarrow C_{\left(x\right)}\ge1\)
Dấu ''='' xảy ra khi \(\left\{{}\begin{matrix}\left(x-5y+7\right)^2=0\\\left(y-3\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-5y+7=0\\y-3=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=8\\y=3\end{matrix}\right.\)
Vậy \(minC_{\left(x\right)}=1\Leftrightarrow x=8;y=3\)
d, \(D_{\left(x\right)}=4x^2-4xy+2y^2-20x-4y+174\)
\(=\left(4x^2+y^2+25-4xy-20x+10y\right)+\left(y-14y+49\right)+74\)
\(=\left(2x-y-5\right)^2+\left(y-7\right)^2+74\ge74\Leftrightarrow D_{\left(x\right)}\ge74\)
Dấu ''='' xảy ra khi \(\left\{{}\begin{matrix}\left(2x-y-5\right)^2=0\\\left(y-7\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}2x-y-5=0\\y-7=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=6\\y=7\end{matrix}\right.\)
Vậy \(minD_{\left(x\right)}=74\Leftrightarrow x=6;y=7\)
e, \(E_{\left(x\right)}=x^2-2x+y^2+4y+5\)
\(=\left(x^2-2x+1\right)+\left(y^2+4y+4\right)=\left(x-1\right)^2+\left(y+2\right)^2\ge0\)
Dấu ''='' xảy ra khi \(\left\{{}\begin{matrix}\left(x-1\right)^2=0\\\left(y+2\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-1=0\\y+2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=1\\y=-2\end{matrix}\right.\)
Vậy \(minE_{\left(x\right)}=0\Leftrightarrow x=1;y=-2\)
bạn ơi! Sao cái chỗ A(x) =(x+y+1)2+(x-2)2-3 mà chuyển sang lại là -3 v
đề thật như vậy sao
K=(4x^2+4xy+y^2)+(x^2-2x+1)+(y^2+4y+4)+2016
=(2x+y)^2+(x-1)^2+(y+2)^2+2016 > =2016 với mọi x,y
minK=2016,dấu "=" xảy ra <=> x=1;y=-2