Em đng cần gấp ạ

B = 2x² + 5x + 7

     = 2( x² + 5/2x + 25/16 ) + 31/8

     = 2( x + 5/4 )² + 31/8

\(2\left(x+\frac{5}{4}\right)^2\ge0\forall x\Rightarrow2\left(x+\frac{5}{4}\right)^2+\frac{31}{8}\ge\frac{31}{8}\)

Đẳng thức xảy ra <=> x + 5/4 => x = -5/4

=> MinB = 31/8 <=> x = -5/4

C = 6x - x² - 12 = -( x² - 6x + 9 ) - 3 = -( x - 3 )² - 3

\(-\left(x-3\right)^2\le0\forall x\Rightarrow-\left(x-3\right)^2-3\le-3\)

Đẳng thức xảy ra <=> x - 3 = 0 => x = 3

=> MaxC = -3 <=> x = 3

D = -3x² - x + 5 = -3( x² + 1/3x + 1/36 ) + 61/12 = -3( x + 1/6 )² + 61/12

\(-3\left(x+\frac{1}{6}\right)^2\le0\forall x\Rightarrow-3\left(x+\frac{1}{6}\right)^2+\frac{61}{12}\le\frac{61}{12}\)

Đẳng thức xảy ra <=> x + 1/6 = 0 => x = -1/6

=> MaxD = 61/12 <=> x = -1/6

a) = 3(x²-2x+1) +1-3

GTNN = -2

B) tt

Giải: Ta có:

B = \(\frac{3x^2-6x+17}{x^2-2x+5}=\frac{3\left(x^2-2x+1\right)+14}{\left(x^2-2x+1\right)+4}=\frac{3\left(x-1\right)^2+14}{\left(x-1\right)^2+4}=3+\frac{14}{\left(x-1\right)^2+4}\)

Do \(\left(x-1\right)^2\ge0\forall x\) => \(\left(x-1\right)^2+4\ge4\forall x\)

 => \(\frac{14}{\left(x-1\right)^2+4}\le\frac{7}{2}\forall x\) 

=> \(3+\frac{14}{\left(x-1\right)^2+4}\le\frac{13}{2}\forall x\)

Dấu "=" xảy ra <=> x - 1 = 0 <=> x = 1

Vậy MaxA = 13/2 <=> x = 1

\(P=-3x^2+6x-y^2+3y+10\)

\(=-3x^2+6x-3-y^2+3y-\frac{9}{4}+\frac{61}{4}\)

\(=-3\left(x^2-2x+1\right)-\left(y^2-3y+\frac{9}{4}\right)+\frac{61}{4}\)

\(=-3\left(x-1\right)^2-\left(y-\frac{3}{2}\right)^2+\frac{61}{4}\)

Vì \(\left(x-1\right)^2\ge0\forall x\)\(\Rightarrow-3\left(x-1\right)^2\le0\forall x\)

\(\left(y-\frac{3}{2}\right)^2\ge0\forall x\)\(\Rightarrow-\left(y-\frac{3}{2}\right)^2\le0\forall y\)

\(\Rightarrow-3\left(x-1\right)^2-\left(y-\frac{3}{2}\right)^2\le0\forall x,y\)

\(\Rightarrow-3\left(x-1\right)^2-\left(y-\frac{3}{2}\right)^2+\frac{61}{4}\le\frac{61}{4}\forall x,y\)

hay \(P\le\frac{61}{4}\)

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

Vậy \(maxP=\frac{61}{4}\)\(\Leftrightarrow\hept{\begin{cases}x=1\\y=\frac{3}{2}\end{cases}}\)

P = -3x² + 6x - y² + 3y + 10

⇔ -P = 3x² - 6x + y² - 3y - 10

          = ( 3x² - 6x + 3 ) + ( y² - 3y + 9/4 ) - 61/4 

          = 3( x² - 2x + 1 ) + ( y - 3/2 )² - 61/4

          = 3( x - 1 )² + ( y - 3/2 )² - 61/4 ≥ -61/4 ∀ x, y

Dấu "=" xảy ra khi x = 1 ; y = 3/2

=> -P ≥ -61/4

=> P ≤ 61/4

=> MaxP = 61/4 ⇔ x = 1 ; y = 3/2

\(E=-3x^2-6x+5\)

\(=-3\left(x^2+2x-\frac{5}{3}\right)\)

\(=-3\left(x^2+2x+1\right)+8\)

\(=-3\left(x+1\right)^2+8\le8\forall x\)

Dau '' = '' xay ra va chi \(\Leftrightarrow\left(x+1\right)^2=0\Leftrightarrow x+1=0\Leftrightarrow x=-1\)

\(E=-3x^2-6x+5=-3\left(x^2+2x+1-1\right)+5\)

\(=-3\left(x+1\right)^2+8\le8\)

Dấu ''='' xảy ra khi x = -1

Vậy GTLN của E bằng 8 tại x = -1 

\(A=x^2-6x+10=\left(x-3\right)^2+1\ge1\)

\(\Rightarrow A_{min}=1\Leftrightarrow x=3\)

\(B=4x^2-4x+25=\left(2x-1\right)^2+24\ge24\)

\(\Rightarrow B_{min}=24\Leftrightarrow x=\frac{1}{2}\)

\(C=3x^2+9x+12=3\left(x+\frac{3}{2}\right)^2+\frac{21}{4}\ge\frac{21}{4}\)

\(\Rightarrow C_{min}=\frac{21}{4}\Leftrightarrow x=\frac{-3}{2}\)

x^2+6x+12=(x+3)^2+3>=3

=>B<=5/3

Dấu = xảy ra khi x=-3