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2.M = 2x2 – 10x + 2y2 + 2xy – 8y + 4038 = (x2 – 10x + 25) +( y2 + 2xy + y2) + ( y2 – 8y + 16) + 3997
= (x-5)2 + (x+y)2 + (y - 4)2 + 3997 = N + 3997
Áp dụng bất đẳng thức Bu- nhi a: (ax+ by + cz)2 \(\le\) (a2+ b2 + c2). (x2 + y2 + z2). Dấu bằng xảy ra khi a/x = b/y = c/z
Ta có: [(5 - x).1 + (x+ y).1 + (y + 4).1]2 \(\le\) [(5 - x)2 + (x+y)2 + (y - 4)2 ].(1+ 1+1) = N .3 = 3.N
<=> 92 = 81 \(\le\) 3.N => N \(\ge\) 27 => 2.M \(\ge\) 27 + 3997 = 4024
=> M \(\ge\)2012
vậy Min M = 2012
khi 5 - x = x+ y = y + 4 => x = 4 ; y = -3
A=x2+y2+xy-5x-4y+2002
2A=x2+2xy+y2+x2-10x+25+y2-8y+16+1961
2A=\(\left(x+y\right)^2+\left(x-5\right)^2+\left(y-4\right)^2+1961\ge1961\)
\(B=\left(x^2+2xy+y^2\right)+\left(x^2-4x+4\right)+2016\)
\(B=\left(x+y\right)^2+\left(y-2\right)^2+2016\)
Vậy Min B =2016 <=> x=-2;y=2
a) \(a+b=2\)
=> \(b=2-a\)
\(A=a^2+\left(2-a\right)^2=2a^2-4a+4=\left(\sqrt{2}a-\sqrt{2}\right)^2+2\ge2\)
Vậy \(A_{min}=2\)
b) \(x+2y=8\)
=> \(x=8-2y\)
\(B=y\left(8-2y\right)=8y-2y^2=8-\left(\sqrt{2}y-2\sqrt{2}\right)^2\le8\)
Vậy \(B_{max}=8\)
a) \(\left(a-b\right)^2\ge0\Leftrightarrow a^2+b^2\ge2ab\Leftrightarrow2\left(a^2+b^2\right)\ge a^2+b^2+2ab\)
\(\Leftrightarrow a^2+b^2\ge\frac{\left(a+b\right)^2}{2}=\frac{2^2}{2}=2\)
Dấu \(=\)khi \(a=b=1\).
b) \(\left(x-2y\right)^2\ge0\Leftrightarrow x^2+4y^2\ge4xy\Leftrightarrow x^2+4xy+4y^2\ge8xy\)
\(\Leftrightarrow xy\le\frac{\left(x+2y\right)^2}{8}=\frac{8^2}{8}=8\)
Dấu \(=\)khi \(\hept{\begin{cases}x=4\\y=2\end{cases}}\).
(3x+4y)² = 9x² + 16y² + 24xy
(4x-3y)² = 16x² + 9y² - 24xy
+ + = + +
=> (3x+4y)² + (4x-3y)² = 25(x²+y²) = 25(14x+6y+6)
=> (3x+4y)² + (4x-3y)² = 2(175x + 75y) + 150
=> (3x+4y)² + (4x-3y)² = 2(99x + 132y + 76x - 57y) + 150
=> (3x+4y)² + (4x-3y)² = 2.33.(3x+4y) + 2.19.(4x-3y) + 150
=> (3x+4y)² - 2.33.(3x+4y) + 33² + (4x-3y)² - 2.19.(4x-3y) + 19² = 150 + 33² + 19²
=> (3x+4y-33)² + (4x-3y-19)² = 1600
Có: (3x+4y-33)² ≤ (3x+4y-33)² + (4x-3y-19)² = 1600
=> 3x+4y-33 ≤ 40 => 3x+4y ≤ 73
max (3x+4y) = 73
đạt khi: 3x+4y = 73 và 4x-3y-19 = 0
=> x = 59/5 và y = 47/5
\(x^2+4y^2-4x+32y+2087=\left(x^2-4x+4\right)+\left(4y^2+32y+64\right)+2019=\left(x-2\right)^2+\left(2y+8\right)^2+2019\ge0+0+2019=2019\Rightarrow B_{min}=2019\Leftrightarrow\left\{{}\begin{matrix}x-2=0\\2y+8=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=-4\end{matrix}\right.\)