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1/
a, \(A=4x^2-4x+5=4x^2-4x+1+4=\left(2x-1\right)^2+4\ge4\)
Dấu "=" xảy ra khi x=1/2
Vậy Amin=4 khi x=1/2
b, \(B=3x^2+6x-1=3\left(x^2+2x+1\right)-4=3\left(x+1\right)^2-4\ge-4\)
Dấu "=" xảy ra khi x=-1
Vậy Bmin = -4 khi x=-1
2/
a, \(A=10+6x-x^2=-\left(x^2-6x+9\right)+19=-\left(x-3\right)^2+19\le19\)
Dấu "=" xảy ra khi x=3
Vậy Amax = 19 khi x=3
b, \(B=7-5x-2x^2=-2\left(x^2-\frac{5}{2}x+\frac{25}{16}\right)+\frac{31}{8}=-2\left(x-\frac{5}{4}\right)^2+\frac{31}{8}\le\frac{31}{8}\)
Dấu "=" xảy ra khi x=5/4
Vậy Bmax = 31/8 khi x=5/4
a) \(A=\left(x+1\right)\left(2x-1\right)\)
\(A=2x^2+x-1\)
\(A=2\left(x^2+\frac{1}{2}x-\frac{1}{2}\right)\)
\(A=2\left[x^2+2\cdot x\cdot\frac{1}{4}+\left(\frac{1}{4}\right)^2-\frac{9}{16}\right]\)
\(A=2\left[\left(x+\frac{1}{4}\right)^2-\frac{9}{16}\right]\)
\(A=2\left(x+\frac{1}{4}\right)^2-\frac{9}{8}\ge\frac{-9}{8}\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x+\frac{1}{4}=0\Leftrightarrow x=\frac{-1}{4}\)
Vậy Amin = -9/8 khi và chỉ khi x = -1/4
b) \(B=4x^2-4xy+2y^2+1\)
\(B=\left(2x\right)^2-2\cdot2x\cdot y+y^2+y^2+1\)
\(B=\left(2x-y\right)^2+y^2+1\ge1\forall x;y\)
Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}2x-y=0\\y=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=0\\y=0\end{cases}\Rightarrow}}x=y=0\)
Vậy Bmin = 1 khi và chỉ khi x = y = 0
bài 1) a) \(A=\left(2x-5\right)^2-4\left(2x-5\right)+5=4x^2-20x+25-8x+20+5\)
\(A=4x^2-28x+49+1=\left(2x-7\right)^2+1\ge1\forall m\)(đpcm)
b) \(A=\left(2x-7\right)^2+1\ge1\) \(\Rightarrow minA=1\Leftrightarrow\left(2x-7\right)^2=0\Leftrightarrow2x-7=0\Leftrightarrow2x=7\Leftrightarrow x=\dfrac{7}{2}\)
Bài 1:
a)\(A=\left(2x-5\right)^2-4\left(2x-5\right)+5\)
\(=\left(2x-5\right)^2-4\left(2x-5\right)+4+1\)
\(=\left(2x-5-2\right)^2+1\)
\(=\left(2x-7\right)^2+1\ge1\)
b)Xảy ra khi \(\left(2x-7\right)^2=0\)
\(\Rightarrow2x-7=0\Rightarrow2x=7\Rightarrow x=\dfrac{7}{2}\)
Bài 2:
a)\(B=-\left(3x+7\right)^2+2\left(3x+7\right)-17\)
\(=-\left(3x+7\right)^2+2\left(3x+7\right)-1-16\)
\(=-\left[\left(3x+7\right)^2-2\left(3x+7\right)+1\right]-16\)
\(=-\left(3x+7-1\right)^2-16\)
\(=-\left(3x+6\right)^2-16\le-16\)
b)Xảy ra khi \(-\left(3x+6\right)^2=0\)
\(\Rightarrow3x+6=0\Rightarrow3x=-6\Rightarrow x=-2\)