Để \(P=\dfrac{2012}{x^2+y^2-20\left(x+y\right)+2213}\) đạt giá trị lớn nhất

\(\Rightarrow x^2+y^2-20\left(x+y\right)+2213\) đạt giá trị nhỏ nhất

\(=x^2-20x+y^2-20x+2213\)

\(=x^2-20x+100+y^2-20y+100+2013\)

\(=\left(x-10\right)^2+\left(y-10\right)^2+2013\ge2013\)

Vậy \(P_{max}=\dfrac{2012}{2013}\) tại \(\left\{{}\begin{matrix}x=10\\y=10\end{matrix}\right.\)

\(\frac{2012}{x^2+y^2-20\left(x+y\right)+2213}=\frac{2012}{\left(x^2-20x+100\right)+\left(y^2-20y+100\right)+2013}\)

\(=\frac{2012}{\left(x-10\right)^2+\left(y-10\right)^2+2013}\le\frac{2012}{2013}\)

\("="\Leftrightarrow x=y=10\)

a: \(\left(x-2\right)^2>=0\)

\(\left|y-x\right|>=0\)

Do đó: \(\left(x-2\right)^2+\left|y-x\right|>=0\forall x,y\)

=>\(\left(x-2\right)^2+\left|y-x\right|+3>=3\forall x,y\)

=>A>=3 với mọi x,y

Dấu = xảy ra khi x-2=0 và y-x=0

=>x=2=y

b: \(\left|x+5\right|>=0\)

=>\(\left|x+5\right|+5>=5\)

=>B>=5 với mọi x

Dấu = xảy ra khi x+5=0

=>x=-5

c: \(\left|x-2010\right|>=0\)

=>\(-\left|x-2010\right|< =0\)

=>\(-\left|x-2010\right|+2012< =2012\)

=>\(C=\dfrac{2011}{2012-\left|x-2010\right|}>=\dfrac{2011}{2012}\forall x\)

Dấu = xảy ra khi x=2010

a) Ta có:

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

Mà: \(\left\{{}\begin{matrix}\left(x-2\right)^2\ge0\\\left|y-x\right|\ge0\end{matrix}\right.\)

\(\Rightarrow A=\left(x-2\right)^2+\left|y-x\right|+3\ge3\)

Dấu "=" xảy ra khi:

\(\left\{{}\begin{matrix}x-2=0\\y-x=0\end{matrix}\right.\)

\(\Rightarrow x=y=2\)

Vậy: \(A_{min}=3\Leftrightarrow x=y=2\) 

b) Ta có:

\(B=\left|x+5\right|+5\)

Mà: \(\left|x+5\right|\ge0\)

\(\Rightarrow B=\left|x+5\right|+5\ge5\)

Dấu "=" xảy ra:

\(x+5=0\Rightarrow x=-5\)

Vậy: \(B_{min}=5\Leftrightarrow x=-5\)

c) Ta có:

\(C=\dfrac{2011}{2012-\left|x-2010\right|}\)

Mà: \(\left|x-2010\right|\ge0\)

\(\Rightarrow C=\dfrac{2011}{2012-\left|x-2010\right|}\ge\dfrac{2011}{2012}\)

Dấu "=" xảy ra khi:

\(x-2010=0\Rightarrow x=2010\)

Vậy: \(C_{min}=\dfrac{2011}{2012}\Leftrightarrow x=2010\)

1: \(y=\frac{2x+m}{x+1}\)

=>y'=\(\frac{\left(2x+m\right)^{\prime}\left(x+1\right)-\left(2x+m\right)\left(x+1\right)^{\prime}}{\left(x+1\right)^2}=\frac{2\left(x+1\right)-\left(2x+m\right)}{\left(x+1\right)^2}=\frac{2-m}{\left(x+1\right)^2}\)
Khi x=0 thì y=\(\frac{2\cdot0+m}{0+1}=\frac{m}{1}=m\)

Khi x=1 thì \(y=\frac{2\cdot1+m}{1+1}=\frac{m+2}{2}\)

TH1: Hàm số đồng biến trên khoảng [0;1]

=>y'>0

=>2-m>0

=>m<2

Khi hàm số đồng biến trên khoảng [0;1] thì y max=y(1)=(m+2)/2

y max=2

=>m+2=4

=>m=2(loại)

TH2: Hàm số đồng biến trên khoảng [0;1]

=>y'<0

=>2-m<0

=>m>2

Khi hàm số nghịch biến trên khoảng [0;1] thì y max=y(0)=m

y max=2

=>m=2(loại)

TH3: Hàm số là hàm hằng

=>2-m=0

=>m=2

Khi m=2 thì \(y=\frac{2\cdot x+2}{x+1}=2\)

=>\(y_{\max}=2\) khi m=2

a: \(5-2\cdot cos^2x\cdot\sin^2x\)

\(=5-2\cdot\left(\sin x\cdot cosx\right)^2\)

\(=5-2\cdot\left(\frac12\cdot\sin2x\right)^2=5-2\cdot\frac14\cdot\sin^22x=-\frac12\cdot\sin^22x+5\)

Ta có: \(0\le\sin^22x\le1\)

=>\(-\frac12\le-\frac12\cdot\sin^22x\le0\)

=>\(-\frac12+5\le-\frac12\cdot\sin^22x+5\le0+5\)

=>\(\frac92\le-\frac12\cdot\sin^22x+5\le5\)

=>\(\frac{3\sqrt2}{2}\le\sqrt{-\frac12\cdot\sin^22x+5}\le\sqrt5\)

=>\(4:\frac{3\sqrt2}{2}\ge\frac{4}{\sqrt{-\frac12\cdot sin^22x+5}}\ge\frac{4}{\sqrt5}\)

=>\(\frac{2\sqrt2}{3}\ge y\ge\frac{4\sqrt5}{5}\)

Do đó: \(y_{\max}=\frac{2\sqrt2}{3}\) khi \(\sin^22x=1\)

=>\(cos^22x=0\)

=>cos2x=0

=>\(2x=\frac{\pi}{2}+k\pi\)

=>\(x=\frac{\pi}{4}+\frac{k\pi}{2}\)

\(y_{\min}=\frac{4\sqrt5}{5}\) khi \(\sin^22x=0\)

=>sin 2x=0

=>\(2x=k\pi\)

=>\(x=\frac{k\pi}{2}\)

b: \(f\left(x\right)=3\cdot\sin^2x+5\cdot cos^2x-4\cdot cos2x-2\)

\(=3\cdot\sin^2x+5\cdot cos^2x-4\left(cos^2x-\sin^2x\right)-2\)

\(=3\cdot\sin^2x+5\cdot cos^2x-4\cdot cos^2x+4\cdot\sin^2x-2\)

\(=7\cdot\sin^2x+cos^2x-2=7\cdot\sin^2x+1-\sin^2x-2=6\cdot\sin^2x-1\)

Ta có: \(0\le\sin^2x\le1\)

=>\(0\le6\sin^2x\le6\)

=>\(0-1\le6\sin^2x-1\le6-1\)

=>-1<=f(x)<=5

f(x) min=-1 khi \(\sin^2x=0\)

=>sin x=0

=>\(x=k\pi\)

f(x) max=5 khi \(\sin^2x=1\)

=>\(cos^2x=0\)

=>cosx=0

=>\(x=\frac{\pi}{2}+k\pi\)

\(\dfrac{1}{y}=\dfrac{x^2+2x+1}{x}=x+\dfrac{1}{x}+2\ge2\sqrt{x.\dfrac{1}{x}}+2=4\)

\(\Rightarrow y\le\dfrac{1}{4}\)

\(y_{max}=4\) khi \(x=1\)