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a, \(A=\dfrac{6}{x^2-2x+3}\)\(=\dfrac{6}{x^2-2x+1+2}=\dfrac{6}{\left(x-1\right)^2+2}\)
Ta có: \(\left(x-1\right)^2\ge0\forall x\Leftrightarrow\left(x-1\right)^2+2\ge2\)
\(\Leftrightarrow\dfrac{1}{\left(x-1\right)^2+2}\le\dfrac{1}{2}\Leftrightarrow\dfrac{6}{\left(x-1\right)^2+2}\le3\)
Dấu bằng xảy ra \(\Leftrightarrow x-1=0\Leftrightarrow x=1\)
Vậy MaxA = 3 khi x = 1
b, \(B=\dfrac{4}{x^2+6x+11}=\dfrac{4}{x^2+6x+9+2}=\dfrac{4}{\left(x+3\right)^2+2}\)
Ta có: \(\left(x+3\right)^2\ge0\forall x\Leftrightarrow\left(x+3\right)^2+2\ge2\)\(\Leftrightarrow\dfrac{1}{\left(x+3\right)^2+2}\le\dfrac{1}{2}\Leftrightarrow\dfrac{4}{\left(x+3\right)^2+2}\le2\)
Dấu bằng xảy ra \(\Leftrightarrow x+3=0\Leftrightarrow x=-3\)
Vậy MaxB = 2 khi x = -3
Bài 2:
\(A=\dfrac{5}{2x-x^2}=\dfrac{5}{-\left(x^2-2x+1\right)+1}=\dfrac{5}{-\left(x-1\right)^2+1}\)
Ta có: \(\left(x-1\right)^2\ge0\forall x\Leftrightarrow-\left(x-1\right)^2\le0\forall x\)
\(\Leftrightarrow-\left(x-1\right)^2+1\le1\Leftrightarrow\dfrac{1}{-\left(x-1\right)^2+1}\ge1\)\(\Leftrightarrow\dfrac{5}{-\left(x-1\right)^2+1}\ge5\)
Dấu bằng xảy ra \(\Leftrightarrow x-1=0\Leftrightarrow x=1\)
Vậy MinA = 5 khi x = 1
Bài 2:
a: Ta có: \(A=\left(x+1\right)^3+\left(x-1\right)^3\)
\(=x^3+3x^2+3x+1+x^3-3x^2+3x-1\)
\(=2x^3+6x\)
b: Ta có: \(B=\left(x-3\right)^3-\left(x+3\right)\left(x^2-3x+9\right)+\left(3x-1\right)\left(3x+1\right)\)
\(=x^3-9x^2+27x-27-x^3-27+9x^2-1\)
\(=27x-55\)
\(a,-x^2+2x+5=-\left(x^2-2x-5\right)=-\left(x^2-2x+1-6\right)=-\left(x-1\right)^2+6\le6\)
dấu'=' xảy ra<=>x=1=>Max A=6
\(b,B=-x^2-y^2+4x+4y+2=-x^2+4x-4-y^2+4x-4+10\)
\(=-\left(x^2-4x+4\right)-\left(y^2-4x+4\right)+10\)
\(=-\left(x-2\right)^2-\left(y-2\right)^2+10=-\left[\left(x-2\right)^2+\left(y-2\right)^2\right]+10\le10\)
dấu"=" xảy ra<=>x=y=2=>Max B=10
\(c,C=x^2+y^2-2x+6y+12=\left(x-1\right)^2+\left(y+3\right)^2+2\ge2\)
dấu'=' xảy ra<=>x=1,y=-3=>MinC=2
Z
Trả lời:
a, \(x^2-6x+11=x^2-6x+9+2=\left(x-3\right)^2+2\ge2\forall x\)
Dấu "=" xảy ra khi x - 3 = 0 <=> x = 3
Vậy GTNN của biểu thức bằng 2 khi x = 3
b, \(-x^2+6x-11=-\left(x^2-6x+11\right)=-\left(x^2-6x+9+2\right)=-\left[\left(x-3\right)^2+2\right]\)
\(=-\left(x-3\right)^2-2\le-2\forall x\)
Dấu "=" xảy ra khi x - 3 = 0 <=> x = 3
Vậy GTLN của biểu thức bằng - 2 khi x = 3
c, \(x^2+2x+2=x^2+2x+1+1=\left(x+1\right)^2+1\ge1>0\forall x\inℤ\) (đpcm)
Dấu "=" xảy ra khi x + 1 = 0 <=> x = - 1
Bài 3:
a) Ta có: \(A=25x^2-20x+7\)
\(=\left(5x\right)^2-2\cdot5x\cdot2+4+3\)
\(=\left(5x-2\right)^2+3>0\forall x\)(đpcm)
d) Ta có: \(D=x^2-2x+2\)
\(=x^2-2x+1+1\)
\(=\left(x-1\right)^2+1>0\forall x\)(đpcm)
Bài 1:
a) Ta có: \(A=x^2-2x+5\)
\(=x^2-2x+1+4\)
\(=\left(x-1\right)^2+4\ge4\forall x\)
Dấu '=' xảy ra khi x=1
b) Ta có: \(B=x^2-x+1\)
\(=x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}\)
\(=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\forall x\)
Dấu '=' xảy ra khi \(x=\dfrac{1}{2}\)
Bạn chú ý đăng lẻ câu hỏi! 1/
a/ \(=x^3-2x^5\)
b/\(=5x^2+5-x^3-x\)
c/ \(=x^3+3x^2-4x-2x^2-6x+8=x^3=x^2-10x+8\)
d/ \(=x^2-x^3+4x-2x+2x^2-8=3x^2-x^3+2x-8\)
e/ \(=x^4-x^2+2x^3-2x\)
f/ \(=\left(6x^2+x-2\right)\left(3-x\right)=17x^2+5x-6-6x^3\)
`A=(5x^2-6x+5)/(x^2-2x+1)`
Xét `A-4`
`=(5x^2-6x+5-4x^2+8x-4)/(x-1)^2`
`=(x^2+2x+1)/(x-1)62`
`=(x+1)^2/(x-1)^2>=0`
`=>A>=4`
Dấu "=" `<=>x+1=0<=>x=-1`
`A=(5x^2-6x+5)/(x^2-2x+1)`
Xét `A-4`
`=(5x^2-6x+5-4x^2+8x-4)/(x-1)^2`
`=(x^2+2x+1)/(x-1)^2`
`=(x+1)^2/(x-1)^2>=0`
`=>A>=4`
Dấu "=" `<=>x+1=0<=>x=-1`
\(A=\frac{6}{x^2-2x+3}=\frac{6}{x^2-2x+1+2}=\frac{6}{\left(x-1\right)^2+2}\le3\)
Dấu = xảy ra khi x-1=0
=> x=1
B tương tự
bài 2:
\(A=\frac{5}{-x^2+2x}=\frac{5}{-\left(x^2-2x+1\right)+1}=\frac{5}{-\left(x-1\right)^2+1}\le5\)(x khác 2)
dấu = xảy ra khi x-1=0
=> x=1
tìm GTLN chứ?????