Bài 8:

\(F=x^2-2x+1+x^2-6x+9=2x^2-8x+10\\ F=2\left(x^2-4x+4\right)+2=2\left(x-2\right)^2+2\ge2\\ F_{min}=2\Leftrightarrow x=2\)

Bài 9:

\(A=-x^2+2x-1+5=-\left(x-1\right)^2+5\le5\\ A_{max}=5\Leftrightarrow x=1\\ B=-x^2+10x-25+2=-\left(x-5\right)^2+2\le2\\ B_{max}=2\Leftrightarrow x=5\\ C=-x^2+6x-9+9=-\left(x-3\right)^2+9\le9\\ C_{max}=9\Leftrightarrow x=3\)

a) Đặt  \(A=-x^2+9x-12\)

\(-A=x^2-9x+12\)

\(-A=\left(x^2-9x+\frac{81}{4}\right)-\frac{33}{4}\)

\(-A=\left(x-\frac{9}{2}\right)^2-\frac{33}{4}\)

Mà  \(\left(x-\frac{9}{2}\right)^2\ge0\forall x\)

\(\Rightarrow-A\ge-\frac{33}{4}\Leftrightarrow A\le\frac{33}{4}\)

Dấu "=" xảy ra khi :  \(x-\frac{9}{2}=0\Leftrightarrow x=\frac{9}{2}\)

Vậy  \(A_{Max}=\frac{33}{4}\Leftrightarrow x=\frac{9}{2}\)

b) Đặt \(B=2x^2+10x-1\)

\(B=2\left(x^2+5x+\frac{25}{4}\right)-\frac{29}{4}\)

\(B=2\left(x+\frac{5}{2}\right)^2-\frac{29}{4}\)

Mà  \(\left(x+\frac{5}{2}\right)^2\ge0\forall x\Rightarrow2\left(x+\frac{5}{2}\right)^2\ge0\forall x\)

\(\Rightarrow B\ge-\frac{29}{4}\)

Dấu "=" xảy ra khi :  \(x+\frac{5}{2}=0\Leftrightarrow x=-\frac{5}{2}\)

Vậy  \(B_{Min}=-\frac{29}{4}\Leftrightarrow x=-\frac{5}{2}\)

c) Đặt  \(C=\left(2x+6\right)\left(x-1\right)\)

\(C=2x^2-2x+6x-6\)

\(C=2x^2+4x-6\)

\(C=2\left(x^2+2x+1\right)-8\)

\(C=2\left(x+1\right)^2-8\)

Mà  \(\left(x+1\right)^2\ge0\forall x\Rightarrow2\left(x+1\right)^2\ge0\forall x\)

\(\Rightarrow C\ge-8\)

Dấu "=" xảy ra khi :  \(x+1=0\Leftrightarrow x=-1\)

Vậy  \(C_{Min}=-8\Leftrightarrow x=-1\)

d) Đặt  \(D=3x-2x^2\)

\(-2D=4x^2-6x\)

\(-2D=\left(4x^2-6x+\frac{9}{4}\right)-\frac{9}{4}\)

\(-2D=\left(2x-\frac{3}{2}\right)^2-\frac{9}{4}\)

Mà  \(\left(2x-\frac{3}{2}\right)^2\ge0\forall x\)

\(\Rightarrow-2D\ge-\frac{9}{4}\)

\(\Leftrightarrow D\le\frac{9}{8}\)

Dấu "=" xảy ra khi :  \(2x-\frac{3}{2}=0\Leftrightarrow x=\frac{3}{4}\)

Vậy  \(D_{Max}=\frac{9}{8}\Leftrightarrow x=\frac{3}{4}\)

\(A=2x^2+4y^2+4xy+10x+12y+18\)

\(A=x^2+4xy+4y^2+6x+12y+9+x^2+4x+4+5\)

\(A=\left(x+2y\right)^2+2.3\left(x+2y\right)+9+\left(x+2\right)^2+5\)

\(A=\left(x+2y+3\right)^2+\left(x+2\right)^2+5\)

Do : \(\hept{\begin{cases}\left(x+2y+3\right)^2\ge0\forall x\\\left(x+2\right)^2\ge0\forall x\end{cases}}\)

\(\Leftrightarrow\left(x+2y+3\right)^2+\left(x+2\right)^2+5\ge5\)

\("="\Leftrightarrow\hept{\begin{cases}x+2y+3=0\\x+2=0\end{cases}\Leftrightarrow\hept{\begin{cases}y=-\frac{1}{2}\\x=-2\end{cases}}}\)

Vậy \(A_{min}=5\Leftrightarrow\hept{\begin{cases}x=-2\\y=-\frac{1}{2}\end{cases}}\)

Chúc bạn học tốt !!!

A = |\(x\) + 5| + 2023

|\(x\) + 5| ≥ 0 ⇒| \(x\) + 5| + 2023 ≥ 2023⇒ A(min) = 2023 xảy ra khi \(x\) = -5

B = (\(x+2\))² - 2023

(\(x\) + 2)2 ≥ 0 ⇒ (\(x\) + 2)2 ≥ - 2023 ⇒ A(min) = -2023  xảy ra khi \(x\) = -2

C = \(x^2\) - 6\(x\) + 20

C = (\(x^2\) - 3\(x\)) - ( 3\(x\) - 9) + 11

C = \(x\)(\(x-3\)) - 3(\(x\) -3) + 11

C = (\(x-3\))(\(x\)-3) + 11

C = (\(x-3\))² + 11

(\(x\) -3)² ≥ 0 ⇒ (\(x\) - 3)² + 11 ≥ 11 vậy C(min) = 11 xảy ra khi \(x=3\)

D = \(x^2\) + 10\(x\) - 25

D = \(x^2\) + 5\(x\) + 5\(x\) + 25 - 55

D = (\(x^2\) + 5\(x\)) + (5\(x\) + 25) - 50

D = \(x\)(\(x\) + 5) + 5(\(x\) + 5)  - 50

D = (\(x\) +5)(\(x\) + 5) - 50

D = ( \(x\) + 5)² - 50

(\(x+5\))² ≥ 0 ⇒ (\(x\) + 5)² - 50 ≥ -50 ⇒ D(min) = -50 xảy ra khi \(x\) = -5

\(y=2+\dfrac{6}{x-3}\)

\(P=3x\left(2+\dfrac{6}{x-3}\right)+2x+2+\dfrac{6}{x-3}\)

\(P=8x+2+\dfrac{18x}{x-3}+\dfrac{6}{x-3}=8x+20+\dfrac{60}{x-3}\)

\(P=8\left(x-3\right)+\dfrac{60}{x-3}+44\ge2\sqrt{\dfrac{480\left(x-3\right)}{x-3}}+44=44+8\sqrt{30}\)

\(P_{min}=44+8\sqrt{30}\) khi \(8\left(x-3\right)=\dfrac{60}{x-3}\Leftrightarrow x=\dfrac{6+\sqrt{30}}{2}\)

Dạ, em cảm ơn thầy ạ

\(A=x^2-4x+1=\left(x^2-4x+4\right)-3=\left(x-2\right)^2-3\ge-3\)

Vậy \(A_{Min}=-3khix=2\)

Bạn có thể giúp mình làm câu còn lại đc ko

mình đang vội lắm

1) \(P=-2x^2-12x=-2\left(x^2+6x+9\right)+18=-2\left(x+3\right)^2+18\le18\)

\(maxP=18\Leftrightarrow x=-3\)

2) \(Q=-5x^2+10x=-5\left(x^2-2x+1\right)+5=-5\left(x-1\right)^2+5\le5\)

\(maxQ=5\Leftrightarrow x=1\)

3) \(A=-3x^2+12x-6=-3\left(x^2-4x+4\right)+6=-3\left(x-2\right)^2+6\le6\)

\(maxA=6\Leftrightarrow x=2\)

4) \(B=-2x^2-24x+12=-2\left(x^2+12x+36\right)+84=-2\left(x+6\right)^2+84\le84\)

\(maxB=84\Leftrightarrow x=-6\)

${Tham\; khảo:(}^{}$

${2x-3)}^2=9$

$\Rightarrow [\begin{array}{c}2x-3=3\\ 2x-3=-3\end{array}\Rightarrow [\begin{array}{c}x=3\\ x=0\end{array}$

Vậy x = 3 hoặc x = 0

\(\left(2x-3\right)^2=9\)

\(\left(2x-3\right)^2=3^2\)

⇒\(2x-3=+-3\)

\(TH1:2x-3=3\text{⇒}x=3\)

\(TH2:2x-3=-3\text{⇒}x=0\)