a) \(A=\sqrt{x-2}+\sqrt{6-x}\)

\(\Rightarrow A^2=x-2+6-x+2\sqrt{\left(x-2\right)\left(6-x\right)}\)

Ta có \(\sqrt{\left(x-2\right)\left(6-x\right)}\ge0,\forall x\)

Do đó \(A^2=4+2\sqrt{\left(x-2\right)\left(6-x\right)}\ge4\)

Mà A không âm \(\Leftrightarrow A\ge2\)

Dấu "=" \(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=6\end{matrix}\right.\)

Áp dụng BĐT Bunhiacopxky:

\(A^2=\left(\sqrt{x-2}+\sqrt{6-x}\right)^2\le\left(x-2+6-x\right)\left(1+1\right)=4\cdot2=8\)

\(\Leftrightarrow A\le\sqrt{8}\)

Dấu "=" \(\Leftrightarrow x-2=6-x\Leftrightarrow x=4\)

Mấy bài còn lại y chang nha 

Tick hộ nha

$A=2x-\sqrt{x}=2(x-\frac{1}{2}\sqrt{x}+\frac{1}{4^2})-\frac{1}{8}$

$=2(\sqrt{x}-\frac{1}{4})^2-\frac{1}{8}$

$\geq \frac{-1}{8}$

Vậy $A_{\min}=-\frac{1}{8}$. Giá trị này đạt tại $x=\frac{1}{16}$

$B=x+\sqrt{x}$

Vì $x\geq 0$ nên $B\geq 0+\sqrt{0}=0$

Vậy $B_{\min}=0$. Giá trị này đạt tại $x=0$

Vì $2-x\geq 0$ (theo ĐKXĐ) nên $C=1+\sqrt{2-x}\geq 1$

Vậy $C_{\min}=1$. Giá trị này đạt tại $2-x=0\Leftrightarrow x=2$

Ta có: \(Q=-x^2-2x+2021\)

\(=-\left(x^2+2x+1-2022\right)\)

\(=-\left(x+1\right)^2+2022\le2022\forall x\)

Dấu '=' xảy ra khi x=-1

\(Q=-\left(x^2+2x+1\right)+2022\)

\(Q=-\left(x+1\right)^2+2022\le2022\)

\(Q_{max}=2022\) khi \(x=-1\)

Em kiểm tra đề là \(\dfrac{y^2}{4}\) hay \(\dfrac{y^4}{4}\)

Nếu đề đúng là \(\dfrac{y^4}{4}\) thì có thể coi như là không giải được

\(2x^2+\dfrac{1}{x^2}+\dfrac{y^2}{4}=4\Leftrightarrow\left(x^2+\dfrac{1}{x^2}-2\right)+\left(x^2-xy+\dfrac{y^2}{4}\right)+xy=2\)

\(\Leftrightarrow2=\left(x-\dfrac{1}{x}\right)^2+\left(x-\dfrac{y}{2}\right)^2+xy\ge xy\)

\(\Rightarrow P_{max}=2023\) khi \(\left\{{}\begin{matrix}x-\dfrac{1}{x}=0\\x-\dfrac{y}{2}=0\end{matrix}\right.\) \(\Rightarrow\left(x;y\right)=\left(-1;-2\right);\left(1;2\right)\)

\(2x^2+\dfrac{1}{x^2}+\dfrac{y^2}{4}=4\Leftrightarrow\left(x^2+\dfrac{1}{x^2}-2\right)+\left(x^2+xy+\dfrac{y^2}{4}\right)-xy=2\)

\(\Rightarrow2=\left(x-\dfrac{1}{x}\right)^2+\left(x+\dfrac{y}{2}\right)^2-xy\ge-xy\)

\(\Rightarrow xy\ge-2\Rightarrow P\ge2019\)

\(P_{min}=2019\) khi \(\left\{{}\begin{matrix}x-\dfrac{1}{x}=0\\x+\dfrac{y}{2}=0\end{matrix}\right.\) \(\Rightarrow\left(x;y\right)=\left(-1;2\right);\left(1;-2\right)\)

tích mình với

ai tích mình

mình tích lại

thanks

Tích mình đi mình tích lại

a, \(x^2+\sqrt{x+2021}=2021\) ĐK \(x\ge-2021\)

<=> \(x^2-2021=-\sqrt{x+2021}\)

Đặt \(\sqrt{x+2021}=a\left(a\ge0\right)\)

=> \(\left\{{}\begin{matrix}x^2-2021=-a\\a^2-2021=x\end{matrix}\right.\)

=> \(\left(x-a\right)\left(x+a\right)+a+x=0\)

<=> \(\left[{}\begin{matrix}x+a=0\\x-a+1=0\end{matrix}\right.\)

+ \(x+a=0\)

=> \(\sqrt{x+2021}=-x\)

=> \(\left\{{}\begin{matrix}x\le0\\x^2-x-2021=0\end{matrix}\right.\)=> \(x=\frac{1-7\sqrt{165}}{2}\)

+ \(x-a+1=0\)

=> \(x+1=\sqrt{x+2021}\)

=> \(\left\{{}\begin{matrix}x\ge-1\\x^2+x-2020\end{matrix}\right.\)=> \(x=\frac{-1+\sqrt{8081}}{2}\)

Vậy \(S=\left\{\frac{-1+\sqrt{8081}}{2};\frac{1-7\sqrt{165}}{2}\right\}\)

a.

\(y'=\dfrac{2-x}{2x^2\sqrt{x-1}}=0\Rightarrow x=2\)

\(y\left(1\right)=0\) ; \(y\left(2\right)=\dfrac{1}{2}\) ; \(y\left(5\right)=\dfrac{2}{5}\)

\(\Rightarrow y_{min}=y\left(1\right)=0\)

\(y_{max}=y\left(2\right)=\dfrac{1}{2}\)

b.

\(y'=\dfrac{1-3x}{\sqrt{\left(x^2+1\right)^3}}< 0\) ; \(\forall x\in\left[1;3\right]\Rightarrow\) hàm nghịch biến trên [1;3]

\(\Rightarrow y_{max}=y\left(1\right)=\dfrac{4}{\sqrt{2}}=2\sqrt{2}\)

\(y_{min}=y\left(3\right)=\dfrac{6}{\sqrt{10}}=\dfrac{3\sqrt{10}}{5}\)

c.

\(y=1-cos^2x-cosx+1=-cos^2x-cosx+2\)

Đặt \(cosx=t\Rightarrow t\in\left[-1;1\right]\)

\(y=f\left(t\right)=-t^2-t+2\)

\(f'\left(t\right)=-2t-1=0\Rightarrow t=-\dfrac{1}{2}\)

\(f\left(-1\right)=2\) ; \(f\left(1\right)=0\) ; \(f\left(-\dfrac{1}{2}\right)=\dfrac{9}{4}\)

\(\Rightarrow y_{min}=0\) ; \(y_{max}=\dfrac{9}{4}\)

d.

Đặt \(sinx=t\Rightarrow t\in\left[-1;1\right]\)

\(y=f\left(t\right)=t^3-3t^2+2\Rightarrow f'\left(t\right)=3t^2-6t=0\Rightarrow\left[{}\begin{matrix}t=0\\t=2\notin\left[-1;1\right]\end{matrix}\right.\)

\(f\left(-1\right)=-2\) ; \(f\left(1\right)=0\) ; \(f\left(0\right)=2\)

\(\Rightarrow y_{min}=-2\) ; \(y_{max}=2\)

\(B=\frac{\left(x+a\right)\left(x+b\right)}{x}=\frac{x^2+x\left(a+b\right)+ab}{x}=x+\frac{ab}{x}+\left(a+b\right)\)

Áp dụng bđt Cauchy : \(x+\frac{ab}{x}\ge2\sqrt{x.\frac{ab}{x}}=2\sqrt{ab}\)

\(\Rightarrow B\ge\left(\sqrt{a}+\sqrt{b}\right)^2\)

Dấu "=" xảy ra khi \(x=\frac{ab}{x}\Rightarrow................\)

Vậy ......................

Bài tìm MAX tồn tại hai giá trị , do k có điều kiện ràng buộc biến x

P=\(\dfrac{10}{2x+\sqrt{x}+2}\) (x\(\ge0\) )

   =\(\dfrac{10}{2\left(x+\dfrac{1}{2}\sqrt{x}+1\right)}=\dfrac{10}{2\left(x+2\dfrac{1}{4}\sqrt{x}+\dfrac{1}{16}+\dfrac{15}{16}\right)}\)     

    =\(\dfrac{10}{2\left(\left(\sqrt{x}\right)^2+2\dfrac{1}{4}+\left(\dfrac{1}{4}\right)^2\right)+\dfrac{15}{8}}=\dfrac{10}{2\left(\sqrt{x}+\dfrac{1}{4}\right)^2+\dfrac{15}{8}}\)    

Do \(2\left(\sqrt{x}+\dfrac{1}{4}\right)^2+\dfrac{15}{8}\ge\dfrac{15}{8}\) \(\Rightarrow\dfrac{10}{2\left(\sqrt{x}+\dfrac{1}{4}\right)^2+\dfrac{15}{8}}\le\dfrac{10}{\dfrac{15}{8}}=\dfrac{16}{3}\)  

Vậy Max P=  \(\dfrac{16}{3}\Leftrightarrow\sqrt{x}+\dfrac{1}{4}=0\Leftrightarrow\sqrt{x}=-\dfrac{1}{4}\) (vô lý)

\(\Rightarrow Ko\)  tồn tại Max P