Ta có:
\(\left|3x-1\right|\ge0\)

Dấu "=" xảy ra \(\Leftrightarrow x=\dfrac{1}{3}\)

\(\left(2y-1\right)^2\ge0\)

Dấu "=" xảy ra \(\Leftrightarrow y=\dfrac{1}{2}\)

\(\Rightarrow\left|3x-1\right|+\left(2y-1\right)^2+2021\ge0\)

Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{1}{2}\end{matrix}\right.\)

Vậy \(A_{min}=2021\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{1}{2}\end{matrix}\right.\)

\(A=\left(x+3\right)^2+2\ge2\\ A_{min}=2\Leftrightarrow x=-3\\ B=\left(x^2+3x+\dfrac{9}{4}\right)-\dfrac{29}{4}=\left(x+\dfrac{3}{2}\right)^2-\dfrac{29}{4}\ge-\dfrac{29}{4}\\ B_{min}=-\dfrac{29}{4}\Leftrightarrow x=-\dfrac{3}{2}\\ C=\left(9x^2-12x+4\right)+2017=\left(3x-2\right)^2+2017\ge2017\\ C_{min}=2017\Leftrightarrow x=\dfrac{2}{3}\)

Lời giải:

PT \(\Leftrightarrow 3x^2+2x(2y-1)+(4y^2+6y+2021-T)=0\)

Coi đây là PT bậc 2 ẩn $x$.

Vì dấu "=" tồn tại nên PT trên luôn có nghiệm

\(\Rightarrow \Delta'=(2y-1)^2-3(4y^2+6y+2021-T)\geq 0\)

\(\Leftrightarrow -8y^2-22y-6062+3T\geq 0\)

\(\Leftrightarrow 3T\geq 8y^2+22y+6062\)

Mà: \(8y^2+22y+6062=8(y+\frac{11}{8})^2+\frac{48375}{8}\geq \frac{48375}{8}\)

\(\Rightarrow T\geq \frac{48375}{8}:3=\frac{16125}{8}\) (đây chính là GTNN của T)

\(\Leftrightarrow \)

\(A=\left|3x+7\right|+\frac{13}{2}\left|3x+7\right|+6\)

Có: \(\left|3x+7\right|\ge0;\frac{13}{2}\left|3x+7\right|\ge0\)

\(\Rightarrow\left|3x+7\right|+\frac{13}{2}\left|3x+7\right|+6\ge6\)

Dấu '=' xảy ra khi: \(\left|3x+7\right|+\frac{13}{2}\left|3x+7\right|=0\)

\(\Leftrightarrow\left|3x+7\right|.\left(\frac{13}{2}+1\right)=0\)

\(\Leftrightarrow\left|3x+7\right|=0\Leftrightarrow3x+7=0\)

 \(\Leftrightarrow x=-\frac{7}{3}\)

Vậy: \(Min_A=6\) tại \(x=-\frac{7}{3}\)

1) \(A=\frac{2018x^2-2.2018x+2018^2}{2018x^2}=\frac{\left(x-2018\right)^2+2017x^2}{2018x^2}=\frac{\left(x-2018\right)^2}{2018x^2}+\frac{2017}{2018}\)

vì \(\frac{\left(x-2018\right)^2}{2018x^2}\ge0\Rightarrow\frac{\left(x-2018\right)^2}{2018x^2}+\frac{2017}{2018}\ge\frac{2017}{2018}\)

dấu = xảy ra khi x-2018=0

=> x=2018

Vậy Min A=\(\frac{2017}{2017}\)khi x=2018

2) \(B=\frac{3x^2+9x+17}{3x^2+9x+7}=\frac{3x^2+9x+7+10}{3x^2+9x+7}=1+\frac{10}{3x^2+9x+7}=1+\frac{10}{3.x^2+9x+7}\)

\(=1+\frac{10}{3.\left(x^2+9x\right)+7}=1+\frac{10}{3.\left[x^2+\frac{2.x.3}{2}+\left(\frac{3}{2}\right)^2\right]-\frac{9}{4}+7}=1+\frac{10}{3.\left(x+\frac{9}{2}\right)^2+\frac{1}{4}}\)

để B lớn nhất => \(3.\left(x+\frac{3}{2}\right)^2+\frac{1}{4}\)nhỏ nhất

mà \(3.\left(x+\frac{3}{2}\right)^2+\frac{1}{4}\ge\frac{1}{4}\)vì \(3.\left(x+\frac{3}{2}\right)^2\ge0\)

dấu = xảy ra khi \(x+\frac{3}{2}=0\)

=> x=\(-\frac{3}{2}\)

Vậy maxB=\(41\)khi x=\(-\frac{3}{2}\)

3) \(M=\frac{3x^2+14}{x^2+4}=\frac{3.\left(x^2+4\right)+2}{x^2+4}=3+\frac{2}{x^2+4}\)

để M lớn nhất => x²+4 nhỏ nhất

mà \(x^2+4\ge4\)(vì x² lớn hơn hoặc bằng 0)

dấu = xảy ra khi x² =0

=> x=0

Vậy Max M\(=\frac{7}{2}\)khi x=0

ps: bài này khá dài, sai sót bỏ qua =))

ê viết lộn dòng này :v

\(MinA=\frac{2017}{2018}\)nha 

Thanks. <3