\(A=x^2+20y^2+8xy-4y+2015\)

\(=\left(x^2+8xy+16y^2\right)+\left(4y^2-4y+1\right)+2014\)

\(=\left(x+4y\right)^2+\left(2y-1\right)^2+2014\ge2014\forall x\)

Dấu "=" xảy ra khi:

\(\hept{\begin{cases}x+4y=0\\2y-1=0\end{cases}\Rightarrow}\hept{\begin{cases}x=-2\\y=\frac{1}{2}\end{cases}}\)

Vậy GTNN của A là 2014 khi \(x=-2,y=\frac{1}{2}\)

\(B=\frac{x^2-2x+2016}{x^2}\)

\(=\frac{2016x^2-2.x.2016+2016^2}{2016x^2}\)

\(=\frac{\left(x^2-2.x.2016+2016^2\right)+2015x^2}{2016x^2}\)

\(=\frac{\left(x-2016\right)^2+2015x^2}{2016x^2}=\frac{\left(x-2016\right)^2}{2016x^2}+\frac{2015}{2016}\ge\frac{2015}{2016}\forall x\)

Dấu "=" xảy ra khi: \(x-2016=0\Rightarrow x=2016\)

Vậy GTNN của B là \(\frac{2015}{2016}\)khi x = 2016

\(P=x^2+20y^2+8xy-4y+2009\)

\(=\left(x^2+8xy+16y^2\right)+\left(4y^2-4y+1\right)+2008\)

\(=\left(x+4y\right)^2+\left(2y-1\right)^2+2008\)

Vì: \(\begin{cases}\left(x+4y\right)^2\ge0\\\left(2y-1\right)^2\ge0\end{cases}\)\(\Rightarrow\left(x+4y\right)^2+\left(2y-1\right)^2\ge0\)

\(\Rightarrow\left(x+4y\right)^2+\left(2y-1\right)^2+2008\ge2008\)

Vậy GTNN của bt trên là 2008 khi \(\begin{cases}x+4y=0\\2y-1=0\end{cases}\)\(\Leftrightarrow\begin{cases}x=-2\\y=\frac{1}{2}\end{cases}\)

\(D=x^2+20y^2+8xy-4y+2009\)

\(\Leftrightarrow D=x^2+16y^2+4y^2+8xy-4y+1+2008\)

\(\Leftrightarrow D=\left(x^2+8xy+16y^2\right)+\left(4y^2-4y+1\right)+2008\)

\(\Leftrightarrow D=\left[x^2+2.x.4y+\left(4y\right)^2\right]+\left[\left(2y\right)^2-2.2y.1+1^2\right]+2008\)

\(\Leftrightarrow D=\left(x+4y\right)^2+\left(2y-1\right)^2+2008\)

Vậy GTNN của \(D=2008\) khi \(\left\{{}\begin{matrix}x+4y=0\\2y-1=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x+4.\left(0,5\right)=0\\y=0,5\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-2\\y=0,5\end{matrix}\right.\)

a) \(C=x^2-4xy+5y^2+10x-22y+28\)

\(\Leftrightarrow C=x^2-4xy+4y^2+y^2+10x-20y-2y+1+25+2\)

\(\Leftrightarrow C=\left(x^2-4xy+4y^2\right)+\left(10x-20y\right)+\left(y^2-2y+1\right)+2+25\)

\(\Leftrightarrow C=\left(x-2y\right)^2+10\left(x-2y\right)+\left(y-1\right)^2+2+25\)

\(\Leftrightarrow C=\left[\left(x-2y\right)^2+10\left(x-2y\right)+25\right]+\left(y-1\right)^2+2\)

\(\Leftrightarrow C=\left(x-2y+5\right)^2+\left(y-1\right)^2+2\)

Vậy GTNN của \(C=2\) khi \(\left\{{}\begin{matrix}x-2y+5=0\\y-1=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x-2.1+5=0\\y=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)

\(A=-x^2+2xy-4y^2+2x+10y-3\)

\(=-x^2+2xy-y^2+2x-2y-1-3y^2+12y-12+10\)

\(=-\left(x^2-2xy+y^2-2x+2y+1\right)-3\left(y^2-4y+4\right)+10\)

\(=-\left(x-y-1\right)^2-3\left(y-2\right)^2+10< =10\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x-y-1=0\\y-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=2\\x=y+1=3\end{matrix}\right.\)

\(B=-4x^2-5y^2+8xy+10y+12\)

\(=-4x^2+8xy-4y^2-y^2+10y-25+37\)

\(=-4\left(x^2-2xy+y^2\right)-\left(y^2-10y+25\right)+37\)

\(=-4\left(x-y\right)^2-\left(y-5\right)^2+37< =37\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x-y=0\\y-5=0\end{matrix}\right.\)

=>x=y=5

\(a,P=\dfrac{x\sqrt{x}+26\sqrt{x}-19-2x-6\sqrt{x}+x-4\sqrt{x}+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\left(x\ge0;x\ne1\right)\\ P=\dfrac{x\sqrt{x}-x+16\sqrt{x}-16}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}=\dfrac{\left(x+16\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\\ P=\dfrac{x+16}{\sqrt{x}+3}\\ b,P=4\Leftrightarrow\dfrac{x+16}{\sqrt{x}+3}=4\\ \Leftrightarrow x+16=4\sqrt{x}+12\\ \Leftrightarrow x-4\sqrt{x}+4=0\Leftrightarrow\left(\sqrt{x}-2\right)^2=0\\ \Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\left(tm\right)\)

\(c,P=\dfrac{x+16}{\sqrt{x}+3}=\dfrac{x-9+25}{\sqrt{x}+3}=\sqrt{x}-3+\dfrac{25}{\sqrt{x}+3}\\ P=\sqrt{x}+3+\dfrac{25}{\sqrt{x}+3}-6\ge2\sqrt{\left(\sqrt{x}+3\right)\cdot\dfrac{25}{\sqrt{x}+3}}-6=2\cdot5-6=4\\ P_{min}=4\Leftrightarrow\left(\sqrt{x}+3\right)^2=25\Leftrightarrow\sqrt{x}+3=5\left(\sqrt{x}+3>0\right)\\ \Leftrightarrow x=4\left(tm\right)\)

\(d,x=3-2\sqrt{2}\Leftrightarrow\sqrt{x}=\sqrt{2}-1\\ \Leftrightarrow P=\dfrac{3-2\sqrt{2}+16}{\sqrt{2}-1+3}=\dfrac{19-2\sqrt{2}}{\sqrt{2}+2}\\ P=\dfrac{\left(19-2\sqrt{2}\right)\left(2-\sqrt{2}\right)}{2}=\dfrac{42-23\sqrt{2}}{2}\)

Lời giải:
Ta có:

$P=2x^2+y^2+2xy+5x+y+\frac{37}{4}$

$=(x^2+y^2+2xy)+x^2+5x+y+\frac{37}{4}$

$=(x+y)^2+(x+y)+(x^2+4x)+\frac{37}{4}$

$=(x+y)^2+(x+y)+\frac{1}{4}+(x^2+4x+4)+5$

$=(x+y+\frac{1}{2})^2+(x+2)^2+5\geq 5$

Vậy $P_{\min}=5$. Giá trị này đạt tại:

$x+y+\frac{1}{2}=x+2=0$

$\Leftrightarrow x=-2; y=\frac{3}{2}$

Lời giải:
Ta có:

$P=2x^2+y^2+2xy+5x+y+\frac{37}{4}$

$=(x^2+y^2+2xy)+x^2+5x+y+\frac{37}{4}$

$=(x+y)^2+(x+y)+(x^2+4x)+\frac{37}{4}$

$=(x+y)^2+(x+y)+\frac{1}{4}+(x^2+4x+4)+5$

$=(x+y+\frac{1}{2})^2+(x+2)^2+5\geq 5$

Vậy $P_{\min}=5$. Giá trị này đạt tại:

$x+y+\frac{1}{2}=x+2=0$

$\Leftrightarrow x=-2; y=\frac{3}{2}$

a)

\(A=4x-x^2+3=-\left(x^2-4x-3\right)=-\left(x^2-4x+4\right)+7=-\left(x-2\right)^2+7\le7\)

Daaus = xayr ra khi: x = 2

b) \(B=4x^2-12x+15=4\left(x^2-3x+9\right)-21=4\left(x-3\right)^2-21\ge-21\)

Dấu = xảy ra khi x = 3

c) \(C=4x^2+2y^2-4xy-4y+1=\left(4x^2-4xy+y^2\right)+\left(y^2-4y+4\right)-3=\left(2x-y\right)^2+\left(y-2\right)^2-3\ge-3\)

Dấu = xảy ra khi

2x = y và y = 2

=> x = 1 và y = 2

a) A = \(-x^2+4x+3=-\left(x-2\right)^2+7\le7\)

Dấu "=" <=> x = 2

b) \(4x^2-12x+15=\left(2x-3\right)^2+6\ge6\)

Dấu "=" xảy ra <=> \(x=\dfrac{3}{2}\)

c) \(4x^2+2y^2-4xy-4y+1\)

= \(\left(4x^2-4xy+y^2\right)+\left(y^2-4y+4\right)-3\)

= \(\left(2x-y\right)^2+\left(y-2\right)^2-3\ge-3\)

Dấu "=" <=> \(\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)

Lời giải:
Đặt $x+7=t$ thì:

$P=(x+8)^4+(x+6)^4=(t+1)^4+(t-1)^4=2t^4+12t^2+2\geq 2, \forall t\in\mathbb{R}$

Do đó $P_{\min}=2$.

Giá trị này đạt tại $t=0\Leftrightarrow x+7=0$

$\Leftrightarrow x=-7$