nè bạn Câu hỏi của Hương Linh - Toán lớp 8 - Học toán với OnlineMath

A = (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 \(\ge\) -36 

Dấu  "="  <=> x = 0 hoặc x = -5 

Vậy A min = -36 <=> x = 0 hoặc x = - 5 .

B=x² - 2x+y² +4y+8

=x²-2x+1+y²+4y+4+3

=(x-1)²+(y+2)²+3

=(x-1)²+(y+2)²+3 \(\ge\)3

Dấu "=" <=>x=1 và y=-2

Vậy A min=3 <=>x=1 và y=-2

3. Tìm giá trị nhỏ nhất của các biểu thứca. A = 4x2  4x 11b. B = (x - 1) (x 2) (x 3) (x 6)c. C = x2 - 2x y2 - 4y 7Ai nha... - Hoc24

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 \(x=\dfrac{1}{2}\)

a: \(A=x^2-2x+1+y^2-4y+4+2=\left(x-1\right)^2+\left(y-2\right)^2+2\ge2\forall x,y\)

Dấu '=' xảy ra khi x=1 và y=2

b: \(B=\left(x^2+5x-6\right)\left(x^2+5x+6\right)\)

\(=\left(x^2+5x\right)^2-36\ge-36\forall x\)

Dấu '=' xảy ra khi x=0 hoặc x=-5

a) \(A=x^2-2x+y^2-4y+7\)

\(=\left(x^2-2x+1\right)+\left(y^2-4y+4\right)+2\)

\(=\left(x-1\right)^2+\left(y-2\right)^2+2\ge2\)

\(A_{min}=2\Leftrightarrow x=1;y=2\).

b) \(B=\left(x-1\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\)

\(=\left(x-1\right)\left(x+6\right)\left(x+2\right)\left(x+3\right)\)

\(=\left(x^2+5x-6\right)\left(x^2+5x+6\right)\)

-Đặt \(t=x^2+5x-6\)

\(B=t\left(t+12\right)=t^2+12t=t^2+12t+36-36=\left(t+6\right)^2-36\ge-36\)

\(B_{min}=-36\Leftrightarrow t=-6\Leftrightarrow x^2+5x-6=-6\Leftrightarrow x^2+5x=0\Leftrightarrow x\left(x+5\right)=0\Leftrightarrow x=0hayx=-5\)

\(A=\left(4x^2+4x+1\right)+10=\left(2x+1\right)^2+10\ge10\)

\(A_{min}=10\) khi \(2x+1=0\Rightarrow x=-\dfrac{1}{2}\)

\(B=\left(x-1\right)\left(x+6\right)\left(x+2\right)\left(x+3\right)=\left(x^2+5x-6\right)\left(x^2+5x+6\right)=\left(x^2+5x\right)^2-36\ge-36\)

\(B_{min}=-36\) khi \(x^2+5x=0\Rightarrow\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)

\(C=\left(x^2-2x+1\right)+\left(y^2-4x+4\right)+2=\left(x-1\right)^2+\left(y-2\right)^2+2\ge2\)

\(C_{min}=2\) khi \(\left(x;y\right)=\left(1;2\right)\)

a. \(A=4x^2+4x+11\)

   \(A=\left(4x^2+4x+1\right)+10\)

  \(A=\left(2x+1\right)^2+10\)

Ta có: \(\left(2x+1\right)^2\ge0;\forall x\) 

\(\Rightarrow A_{min}=10\)

Dấu "=" xảy ra khi \(\left(2x+1\right)^2=0\)

                            \(\Leftrightarrow2x+1=0\Leftrightarrow x=-\dfrac{1}{2}\)

c.\(C=x^2-2x+y^2-4y+7\)

  \(C=\left(x^2-2x+1\right)+\left(y^2-4y+4\right)+2\)

  \(C=\left(x-1\right)^2+\left(y-2\right)^2+2\)

Ta có: \(\left(x-1\right)^2\ge0;\left(y-2\right)^2\ge0;\forall x,y\)

\(\Rightarrow C_{min}=2\)

Dấu "=" xảy ra khi\(\left\{{}\begin{matrix}\left(x-1\right)^2=0\\\left(y-2\right)^2=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\y-2=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)

`x^2+x+1=x^2+x+1/4+3/4=(x+1/2)^2 +3/4`

Vì `(x+1/2)^2 >= 0` với mọi `x`

  `=>(x+1/2)^2 +3/4 >= 3/4` với mọi `x`

 `=>` Biểu thức Min `=3/4<=>x=-1/2`

_____________

`(x-3)(x+5)+4=x^2+2x-11=x^2+2x+1-12=(x+1)^2-12`

  Vì `(x+1)^2 >= 0` với mọi `x`

    `=>(x+1)^2-12 >= -12` với mọi `x`

 `=>` Biểu thức Min `=-1/2<=>x=-1`

\(a,\\ A=25x^2-10x+11\\ =\left(5x\right)^2-2.5x.1+1^2+10\\ =\left(5x+1\right)^2+10\ge10\forall x\in R\\ Vậy:min_A=10.khi.5x+1=0\Leftrightarrow x=-\dfrac{1}{5}\\ B=\left(x-3\right)^2+\left(11-x\right)^2\\ =\left(x^2-6x+9\right)+\left(121-22x+x^2\right)\\ =x^2+x^2-6x-22x+9+121=2x^2-28x+130\\ =2\left(x^2-14x+49\right)+32\\ =2\left(x-7\right)^2+32\\ Vì:2\left(x-7\right)^2\ge0\forall x\in R\\ Nên:2\left(x-7\right)^2+32\ge32\forall x\in R\\ Vậy:min_B=32.khi.\left(x-7\right)=0\Leftrightarrow x=7\\Tương.tự.cho.biểu.thức.C\)

b:

\(D=-25x^2+10x-1-10\)

\(=-\left(25x^2-10x+1\right)-10\)

\(=-\left(5x-1\right)^2-10< =-10\)

Dấu = xảy ra khi x=1/5

\(E=-9x^2-6x-1+20\)

\(=-\left(9x^2+6x+1\right)+20\)

\(=-\left(3x+1\right)^2+20< =20\)

Dấu = xảy ra khi x=-1/3

\(F=-x^2+2x-1+1\)

\(=-\left(x^2-2x+1\right)+1=-\left(x-1\right)^2+1< =1\)

Dấu = xảy ra khi x=1

Bài 2 : 

a, \(x^2-4x+4+1=\left(x-2\right)^2+1\ge1\)

Dấu ''='' xảy ra khi x = 2 

b, Ta có \(\left(x+1\right)^2+10\ge10\Rightarrow\dfrac{-100}{\left(x+1\right)^2+10}\ge-\dfrac{100}{10}=-10\)

Dấu ''='' xảy ra khi x = -1 

 Bài 1 : 

a, Ta có \(A\left(x\right)=x^2-4x+4=0\Leftrightarrow\left(x-2\right)^2=0\Leftrightarrow x=2\)

b, \(B\left(x\right)=x^2\left(2x+1\right)+\left(2x+1\right)=\left(x^2+1>0\right)\left(2x+1\right)=0\Leftrightarrow x=-\dfrac{1}{2}\)

c, \(C\left(x\right)=\left|2x-3\right|=\dfrac{1}{3}\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{1}{3}+3=\dfrac{10}{3}\\2x=-\dfrac{1}{3}+3=\dfrac{8}{3}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5}{3}\\x=\dfrac{4}{3}\end{matrix}\right.\)