1) Tìm Min A= 12x^2 - 6x + 4 / x^2 + 1
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VM
1
9 tháng 7 2023
Bài 1 :
\(A=-x^2+6x+14\)
\(A=-x^2+6x-9+23\)
\(A=-\left(x^2-6x+9\right)+23\)
\(A=-\left(x-3\right)^2+23\)
Vì \(-\left(x-3\right)^2\le0\)
\(\Rightarrow A=-\left(x-3\right)^2+23\le23\)
\(\Rightarrow Max\left(A\right)=23\)
Bài 2 :
\(B=4x^2+12x+30\)
\(\Rightarrow B=4x^2+12x+9+21\)
\(\Rightarrow B=\left(2x+3\right)^2+21\)
Vì \(\left(2x+3\right)^2\ge0\)
\(\Rightarrow B=\left(2x+3\right)^2+21\ge21\)
\(\Rightarrow Min\left(B\right)=21\)
H9
HT.Phong (9A5)
CTVHS
31 tháng 7 2023
1) \(8x^3-12x^2+6x-1=0\)
\(\Leftrightarrow\left(2x\right)^2-3\cdot\left(2x\right)^2\cdot1+3\cdot2x\cdot1^2-1^3=0\)
\(\Leftrightarrow\left(2x-1\right)^3=0\)
\(\Leftrightarrow2x-1=0\)
\(\Leftrightarrow2x=1\)
\(\Leftrightarrow x=\dfrac{1}{2}\)
2) \(x^3-6x^2+12x-8=27\)
\(\Leftrightarrow x^3-3\cdot x^2\cdot2+3\cdot2^2\cdot x-2^3=27\)
\(\Leftrightarrow\left(x-2\right)^3=27\)
\(\Leftrightarrow\left(x-2\right)^3=3^3\)
\(\Leftrightarrow x-2=3\)
\(\Leftrightarrow x=3+2\)
\(\Leftrightarrow x=5\)
3) \(x^2-8x+16=5\left(4-x\right)^3\)
\(\Leftrightarrow\left(x-4\right)^2=5\left(4-x\right)^3\)
\(\Leftrightarrow\left(4-x\right)^2=5\left(4-x\right)^3\)
\(\Leftrightarrow5\left(4-x\right)=1\)
\(\Leftrightarrow4-x=\dfrac{1}{5}\)
\(\Leftrightarrow x=4-\dfrac{1}{5}\)
\(\Leftrightarrow x=\dfrac{19}{5}\)
4) \(\left(2-x\right)^3=6x\left(x-2\right)\)
\(\Leftrightarrow8-12x+6x^2-x^3=6x^2-12x\)
\(\Leftrightarrow-12x+6x^2-6x^2+12x=8-x^3\)
\(\Leftrightarrow8-x^3=0\)
\(\Leftrightarrow x^3=8\)
\(\Leftrightarrow x^3=2^3\)
\(\Leftrightarrow x=2\)
5) \(\left(x+1\right)^3-\left(x-1\right)^3-6\left(x-1\right)^2=-10\)
\(\Leftrightarrow x^3+3x^2+3x+1-x^3+3x^2-3x+1-6\left(x^2-2x+1\right)=-10\)
\(\Leftrightarrow\left(x^3-x^3\right)+\left(3x-3x\right)+\left(3x^2+3x^2\right)+\left(1+1\right)-6x^2+12x-6=-10\)
\(\Leftrightarrow0+0+0+\left(6x^2-6x^2\right)+12x-4=-10\)
\(\Leftrightarrow12x-4=-10\)
\(\Leftrightarrow12x=-10+4\)
\(\Leftrightarrow12x=-6\)
\(\Leftrightarrow x=\dfrac{-6}{12}\)
\(\Leftrightarrow x=-\dfrac{1}{2}\)
6) \(\left(3-x\right)^3-\left(x+3\right)^3=36x^2-54x\)
\(\Leftrightarrow27-27x+9x^2-x^3-x^3-9x^2-27x-27=36x^2-54x\)
\(\Leftrightarrow-54x-2x^3=36x^2-54x\)
\(\Leftrightarrow-2x^3=36x^2\)
\(\Leftrightarrow-2x^3-36x^2=0\)
\(\Leftrightarrow-2x^2\left(x+18\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}-2x^2=0\\x+18=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-18\end{matrix}\right.\)
TD
2
26 tháng 6 2018
\(A=4x^2-12x+11\)
\(A=\left(2x\right)^2-2.2x.3+3^2+2\)
\(A=\left(2x-3\right)^2+2\)
Ta có: \(\left(2x-3\right)^2\ge0\forall x\)
\(\Rightarrow\left(2x-3\right)^2+2\ge2\forall x\)
Dấu = xảy ra \(\Leftrightarrow\left(2x-3\right)^2=0\Leftrightarrow2x-3=0\Leftrightarrow2x=3\Leftrightarrow x=\frac{3}{2}\)
Vậy Amin=2\(\Leftrightarrow x=\frac{3}{2}\)
\(B=x^2-2x+y^2+4y+6\)
\(B=\left(x^2-2x+1\right)+\left(y^2+2.2y+2^2\right)+1\)
\(B=\left(x-1\right)^2+\left(y+2\right)^2+1\)
Ta có: \(\hept{\begin{cases}\left(x-1\right)^2\ge0\forall x\\\left(y+2\right)^2\ge0\forall y\end{cases}\Rightarrow\left(x-1\right)^2+\left(y+2\right)^2+1\ge1\forall x;y}\)
Dấu = xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(x-1\right)^2=0\\\left(y+2\right)^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x-1=0\\y+2=0\end{cases}\Leftrightarrow}\hept{\begin{cases}x=1\\y=-2\end{cases}}}\)
Vậy Bmin=1\(\Leftrightarrow x=1;y=-2\)
\(A=-x^2-6x+1\)
\(\Rightarrow-A=x^2+6x-1\)
\(-A=\left(x^2+2.3x+3^2\right)-10\)
\(-A=\left(x+3\right)^2-10\)
\(\Rightarrow A=-\left(x+3\right)^2+10\)
Ta có: \(\left(x+3\right)^2\ge0\forall x\Rightarrow-\left(x+3\right)^2\le0\forall x\Rightarrow-\left(x+3\right)^2+10\le10\forall x\)
Dấu = xảy ra \(\Leftrightarrow-\left(x+3\right)^2=0\Leftrightarrow\left(x+3\right)^2=0\Leftrightarrow x+3=0\Leftrightarrow x=-3\)
Vậy Amax=10\(\Leftrightarrow\)x= -3
Sửa đề:
\(B=-2x^2-8x-6\)
\(B=-2.\left(x^2+2.2x+2^2\right)+2\)
\(B=-2.\left(x+2\right)^2+2\)
Ta có: \(2.\left(x+2\right)^2\ge0\forall x\Rightarrow-2.\left(x+2\right)^2\le0\forall x\Rightarrow-2.\left(x+2\right)^2+2\le2\forall x\)
Dấu = xảy ra \(\Leftrightarrow-2.\left(x+2\right)^2=0\Leftrightarrow\left(x+2\right)^2=0\Leftrightarrow x+2=0\Leftrightarrow x=-2\)
Vậy Bmax=2\(\Leftrightarrow x=-2\)
S
26 tháng 6 2018
Đề phải là tìm min mới đúng
a, A=4x2-12x+11
=(4x2-12x+9)+2
=(2x-3)2+2
Vì (2x-3)2 \(\ge\) 0 => A=(2x-3)2+2 \(\ge\) 2
Dấu "=" xảy ra khi 2x-3=0 <=> x=3/2
Vậy Amin = 2 khi x=3/2
b, B=x2-2x+y2+4y+6
=(x2-2x+1)+(y2+4y+4)+1
=(x-1)2+(y+2)2+1
Vì \(\left(x-1\right)^2\ge0;\left(y+2\right)^2\ge0\)
\(\Rightarrow\left(x-1\right)^2+\left(y+2\right)^2\ge0\)
\(\Rightarrow B=\left(x-1\right)^2+\left(y+2\right)^2+1\ge1\)
Dấu "=" xảy ra khi x=1,y=-2
Vậy Bmin = 1 khi x=1,y=-2
HK
3
11 tháng 8 2016
\(b,\left(x^2-9\right)^2=12x+1\)
\(\Leftrightarrow x^4-18x^2+81-12x-1=0\)
\(\Leftrightarrow x^4-18x^2-12x+80=0\)
\(\Leftrightarrow x^4-2x^3+2x^3-4x^2-14x^2+28x-40x+80=0\)
\(\Leftrightarrow x^3\left(x-2\right)+2x^2\left(x-2\right)-14x\left(x-2\right)-40\left(x-2\right)=0\)
\(\Leftrightarrow\left(x^3+2x^2-14x-40\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left(x^3-4x^2+6x^2-24x+10-40\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left[x^2\left(x-4\right)+6x\left(x-4\right)+10\left(x-4\right)\right]\left(x-2\right)=0\)
\(\Leftrightarrow\left(x^2+6x+10\right)\left(x-4\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left[\left(x+3\right)^2+1\right]\left(x-4\right)\left(x-2\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x-4=0\\x-2=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=4\\x=2\end{cases}}\)
Vậy \(S=\left\{2;4\right\}\)
11 tháng 8 2016
\(a,x^4+x^2+6x-8=0\Leftrightarrow x^4+2x^2+1-x^2+6x-9=0\)
\(\Leftrightarrow\left(x^2+1\right)^2-\left(x-3\right)^2=0\Leftrightarrow\left(x^2+x-2\right)\left(x^2-x+4\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x+2\right)\left[\left(x+\frac{1}{2}\right)^2+\frac{15}{4}\right]=0\)
\(\Leftrightarrow\orbr{\begin{cases}x-1=0\\x+2=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=1\\x=-2\end{cases}}}\)
Vậy \(S=\left\{1;-2\right\}\)
S
1
CD
8 tháng 12 2021
a, <=> x2 -2x +1 + 5x -x2 =8
<=> 3x +1 =8
<=> 3x = 7
<=> x= 7/3
b, thiếu đề
c, <=> 2x3 -1 + 2x(4 -x2) = 7
<=> 2x3 + 8x -23 = 8
<=> 8x =8
<=> x=1
TM
1
TN
5 tháng 6 2016
-12x^2 (5x+4) +6x * (10x^2+8x)-2x(x+1)= -4
\(VT=-2x^2-2x\)
\(\Leftrightarrow-2x^2-2x=-4\)
\(\Leftrightarrow-2x^2-2x+4=0\)
\(\Leftrightarrow-2\left(x^2+x-2\right)=0\)
\(\Leftrightarrow x^2+x-2=0\)
\(\Leftrightarrow x^2-x+2x-2=0\)
\(\Leftrightarrow x\left(x-1\right)+2\left(x-1\right)=0\)
\(\Leftrightarrow\left(x+2\right)\left(x-1\right)=0\)
\(\Leftrightarrow x=-2\)hoặc\(x=1\)
11 tháng 10 2020
A = 2x2 + 6x = 2( x2 + 3x + 9/4 ) - 9/2 = 2( x + 3/2 )2 - 9/2 ≥ -9/2 ∀ x
Dấu "=" xảy ra khi x = -3/2
=> MinA = -9/2 <=> x = -3/2
B = x2 - 2x + y2 - 4y + 6 = ( x2 - 2x + 1 ) + ( y2 - 4y + 4 ) + 1 = ( x - 1 )2 + ( y - 2 )2 + 1 ≥ 1 ∀ x, y
Dấu "=" xảy ra khi x = 1 ; y = 2
=> MinB = 1 <=> x = 1 ; y = 2
C = x2 - 2xy + 6y2 - 12x + 2y + 45
= ( x2 - 2xy + y2 - 12x + 12y + 36 ) + ( 5y2 - 10y + 5 ) + 4
= [ ( x2 - 2xy + y2 ) - ( 12x - 12y ) + 36 ] + 5( y2 - 2y + 1 ) + 4
= [ ( x - y )2 - 2( x - y ).6 + 62 ] + 5( y - 1 )2 + 4
= ( x - y - 6 )2 + 5( y - 1 )2 + 4 ≥ 4 ∀ x, y
Dấu "=" xảy ra khi x = 7 ; y = 1
=> MinC = 4 <=> x = 7 ; y = 1
D = ( x - 1 )( x + 2 )( x + 3 )( x + 6 )
= [ ( x - 1 )( x + 6 ) ][ ( x + 2 )( x + 3 ) ]
= ( x2 + 5x - 6 )( x2 + 5x + 6 )
= ( x2 + 5x )2 - 36 ≥ -36 ∀ x
Dấu "=" xảy ra <=> x2 + 5x = 0
<=> x( x + 5 ) = 0
<=> x = 0 hoặc x = -5
=> MinD = -36 <=> x = 0 hoặc x = -5
Ta có A=12x2-6x+4/x2+1
A=(9x2-6x+1)+(3x2+3)/x2+1
A=(3x-1)2+3(x2+1)/x2+1
A= ( (3x-1)2/x2+1 ) +( 3(x2+1)/x2+1 )
A= ( (3x-1)2/x2+1 ) +3
Ta thấy (3x-1)2/x2+1 >= 0 với mọi x
Suy ra A>= 3
Dấu "=" xảy ra khi và chỉ khi (3x-1)2/x2+1 =0
<=> (3x-1)2=0
x =1/3