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g) G = x2 + 6x + 4y2 - 10y + 5
G = (x2+ 6x + 9) + 4(y2 - 2,5y + 1,5625) - 10,25
G = (x + 3)2 + 4(y - 1,25)2 - 10,25 \(\ge\)-10,25 với mọi x;y
Dấu "=" xảy ra <=> \(\hept{\begin{cases}x+3=0\\y-1,25=0\end{cases}}\) <=> \(\hept{\begin{cases}x=-3\\y=1,25\end{cases}}\)
Vậy MinG = -10,25 khi x = -3 và y = 1,25
h) H = -2x2 - 6x - 3y2 + 12y - 8
H = -2(x2 + 3x + 2,25) - 3(y2 - 4y + 4)+ 8,5
H = -2(x + 1,5)2 - 3(Y - 2)2 + 8,5 \(\le\)8,5 với mọi x;y
Dấu "=" xảy ra <=> \(\hept{\begin{cases}x+1,5=0\\y-2=0\end{cases}}\)<=> \(\hept{\begin{cases}x=-1,5\\y=2\end{cases}}\)
vậy MaxH = 8,5 khi x = -1,5 và y = 2
`@` `\text {Ans}`
`\downarrow`
`a)`
`3x(4x-1) - 2x(6x-3) = 30`
`=> 12x^2 - 3x - 12x^2 + 6x = 30`
`=> 3x = 30`
`=> x = 30 \div 3`
`=> x=10`
Vậy, `x=10`
`b)`
`2x(3-2x) + 2x(2x-1) = 15`
`=> 6x- 4x^2 + 4x^2 - 2x = 15`
`=> 4x = 15`
`=> x = 15/4`
Vậy, `x=15/4`
`c)`
`(5x-2)(4x-1) + (10x+3)(2x-1) = 1`
`=> 5x(4x-1) - 2(4x-1) + 10x(2x-1) + 3(2x-1)=1`
`=> 20x^2-5x - 8x + 2 + 20x^2 - 10x +6x - 3 =1`
`=> 40x^2 -17x - 1 = 1`
`d)`
`(x+2)(x+2)-(x-3)(x+1)=9`
`=> x^2 + 2x + 2x + 4 - x^2 - x + 3x + 3=9`
`=> 6x + 7 =9`
`=> 6x = 2`
`=> x=2/6 =1/3`
Vậy, `x=1/3`
`e)`
`(4x+1)(6x-3) = 7 + (3x-2)(8x+9)`
`=> 24x^2 - 12x + 6x - 3 = 7 + (3x-2)(8x+9)`
`=> 24x^2 - 12x + 6x - 3 = 7 + 24x^2 +11x - 18`
`=> 24x^2 - 6x - 3 = 24x^2 + 18x -11`
`=> 24x^2 - 6x - 3 - 24x^2 + 18x + 11 = 0`
`=> 12x +8 = 0`
`=> 12x = -8`
`=> x= -8/12 = -2/3`
Vậy, `x=-2/3`
`g)`
`(10x+2)(4x- 1)- (8x -3)(5x+2) =14`
`=> 40x^2 - 10x + 8x - 2 - 40x^2 - 16x + 15x + 6 = 14`
`=> -3x + 4 =14`
`=> -3x = 10`
`=> x= - 10/3`
Vậy, `x=-10/3`
\(K=\frac{-7}{-2x^2+8x-60}\)
\(K=\frac{-7}{-2\left(x^2-4x+4-26\right)}\)
\(K=\frac{7}{2\left(x-2\right)^2-56}\)
Ta có : \(2\left(x-2\right)^2-56\ge-56\)
\(\Rightarrow K_{max}=\frac{-7}{56}\Leftrightarrow x=2\)
\(L=\frac{8}{-3x^2+9x-40}\)
\(L=\frac{8}{-3\left(x^2-3x+\frac{9}{4}+\frac{133}{12}\right)}\)
\(L=\frac{-8}{3\left(x-\frac{3}{2}\right)^2+\frac{133}{4}}\)
Ta có : \(3\left(x-\frac{3}{2}\right)^2+\frac{133}{4}\ge\frac{133}{4}\)
\(\Rightarrow L_{max}=-\frac{8.4}{133}=-\frac{32}{133}\Leftrightarrow x=\frac{3}{2}\)
a. Ta có : \(A=\frac{8x^2-9}{x^2+3}=\frac{8x^2+24-33}{x^2+3}=8-\frac{33}{x^2+3}\)
Để Amin thì \(\frac{33}{x^2+3}_{max}\) mà \(\frac{33}{x^2+3}\le11\)
Dấu "=" xảy ra \(\Leftrightarrow x^2+3=3\Leftrightarrow x=0\)
Vậy Amin = 8 - 11 = - 3 <=> x = 0
b. Ta có : \(B=\frac{3x^2-6x+40}{x^2-2x+5}=\frac{3\left(x^2-2x+5\right)+25}{x^2-2x+5}=3+\frac{25}{x^2-2x+5}\)
Để Bmax thì \(\frac{25}{x^2-2x+5}=\frac{25}{\left(x-1\right)^2+4}_{max}\)
mà \(\frac{25}{\left(x-1\right)^2+4}\le\frac{25}{4}\)
Dấu "=" xảy ra \(\Leftrightarrow\left(x-1\right)^2+4=4\Leftrightarrow x-1=0\Leftrightarrow x=1\)
Vậy Bmax \(=3+\frac{25}{4}=\frac{37}{4}\) <=> x = 1
a/ Ta có:
\(A=x^2-6x+11\)
\(A=x\cdot x-3x-3x+3\cdot3+2\)
\(A=x\left(x-3\right)-3\left(x-3\right)+2\)
\(A=\left(x-3\right)\left(x-3\right)+2\)
\(A=\left(x-3\right)^2+2\)
Vì \(\left(x-3\right)^2\ge0\)
Nên GTNN của \(\left(x-3\right)^2\)là 0
=> \(A_{min}=0+2=2\)
mình chỉ biết a. thôi
a) ta có : \(A=x^2-6x+11\)
\(A=x.x-3x-3x+3.3+2\)
\(A=x\left(x-3\right)-3\left(x-3\right)+2\)
\(A=\left(x-3\right)\left(x-3\right)+2\)
\(A=\left(x-3\right)^2+2\)
vì \(\left(x-3\right)^2\ge0\)
nên GTNN của \(\left(x-3\right)^2\)là \(0\)
\(\Rightarrow\)\(A_{min}\)\(=0+2=2\)
\(A=x^2-4x+10=x^2-4x+4+6=\left(x-2\right)^2+6\ge6\)
Vậy GTNN A là 6 khi x - 2 = 0 <=> x = 2
\(B=\left(1-x\right)\left(3x-4\right)=3x-4-3x^2+4x=-3x^2+7x-4\)
\(=-3\left(x^2-\frac{7}{3}x+\frac{4}{3}\right)=-3\left(x^2-2.\frac{7}{6}x+\frac{49}{36}-\frac{1}{36}\right)=-3\left(x-\frac{7}{6}\right)^2+\frac{1}{12}\ge\frac{1}{12}\)
\(=3\left(x-\frac{7}{6}\right)^2-\frac{1}{12}\le-\frac{1}{12}\)Vậy GTLN B là -1/12 khi x = 7/6
\(C=3x^2-9x+5=3\left(x^2-3x+\frac{5}{3}\right)=3\left(x^2-2.\frac{3}{2}x+\frac{9}{4}-\frac{7}{12}\right)\)
\(=3\left(x-\frac{3}{2}\right)^2-\frac{7}{4}\ge-\frac{7}{4}\)Vậy GTNN C là -7/4 khi x = 3/2
\(D=-2x^2+5x+2=-2\left(x^2-\frac{5}{2}x-1\right)=-2\left(x^2-2.\frac{5}{4}x+\frac{25}{16}-\frac{41}{16}\right)\)
\(=-2\left(x-\frac{5}{4}\right)^2+\frac{21}{8}\le\frac{21}{8}\)Vậy GTLN D là 21/8 khi x = 5/4