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11 tháng 6 2021

\(A=\)\(\sqrt{x+2\left(1+\sqrt{x+1}\right)}+\sqrt{x+2\left(1-\sqrt{x+1}\right)}\) (đk: \(x\ge-1\))

\(=\sqrt{\left(x+1\right)+2\sqrt{x+1}+1}+\sqrt{\left(x+1\right)-2\sqrt{x+1}+1}\)

\(=\sqrt{\left(\sqrt{x+1}+1\right)^2}+\sqrt{\left(\sqrt{x+1}-1\right)^2}\)

\(=\sqrt{x+1}+1+\left|\sqrt{x+1}-1\right|\)

\(=\left[{}\begin{matrix}\sqrt{x+1}+1+\sqrt{x+1}-1;\sqrt{x+1}\ge1\\\sqrt{x+1}+1-\left(\sqrt{x+1}-1\right);\sqrt{x+1}< 1\end{matrix}\right.\)

\(=\left[{}\begin{matrix}2\sqrt{x+1};x\ge0\\2;-1\le x< 0\end{matrix}\right.\)

Có \(2\sqrt{x+1}\ge2\) tại \(x\ge0\) 

\(\Rightarrow\min\limits_{x\ge0}A=2\)

Dấu = xảy ra <=> x=0 mà tại \(-1\le x< 0\) thì A=2

Vậy giá trị nhỏ nhất của biểu thức là 2 tại x=0 hoặc \(-1\le x< 0\)

(Ủa đề zì kì)

\(ĐKXĐ:x\ge-1\)

Đặt \(A=\sqrt{x+2\left(1+\sqrt{x+1}\right)}+\sqrt{x+2\left(1-\sqrt{x+1}\right)}\)

\(=\sqrt{x+1+2\sqrt{x+1}+1}+\sqrt{x+1-2\sqrt{x+1}+1}\)

\(=\sqrt{\left(\sqrt{x+1}+1\right)^2}+\sqrt{\left(\sqrt{x+1}-1\right)^2}\)

\(=\left|\sqrt{x+1}+1\right|+\left|\sqrt{x+1}-1\right|\)

\(=\left|\sqrt{x+1}+1\right|+\left|1-\sqrt{x+1}\right|\)

\(\ge\left|\sqrt{x+1}+1+1-\sqrt{x+1}\right|=2\)

Dấu "=" xảy ra khi \(\left(\sqrt{x+1}+1\right)\left(1-\sqrt{x+1}\right)\ge0\)

\(\Leftrightarrow1-\sqrt{x+1}\ge0\)

\(\Leftrightarrow\sqrt{x+1}\le1\)

\(\Leftrightarrow x\le0\). Mà \(x\ge-1\) Nên \(-1\le x\le0\)

Vậy Min \(A=2\) khi \(-1\le x\le0\)

29 tháng 12 2020

\(ĐKXĐ:\left\{{}\begin{matrix}x>0\\x\ne1\end{matrix}\right.\)

\(P\left(x\right)=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+\dfrac{2\left(x-1\right)}{\sqrt{x}-1}\)

\(P\left(x\right)=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\left(x+\sqrt{x}+1\right)}-\dfrac{\sqrt{x}\left(2\sqrt{x}+1\right)}{\sqrt{x}}+\dfrac{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}-1}\)

\(P\left(x\right)=x-\sqrt{x}-2\sqrt{x}-2+2\sqrt{x}+2\)

\(P\left(x\right)=x-\sqrt{x}\)

Ta có : \(\dfrac{P\left(x\right)}{2020\sqrt{x}}=\dfrac{x-\sqrt{x}}{2020\sqrt{x}}=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{2020\sqrt{x}}=\dfrac{\sqrt{x}-1}{2020}\)

Để \(\dfrac{P\left(x\right)}{2020\sqrt{x}}min\Leftrightarrow\dfrac{\sqrt{x}-1}{2020}min\Leftrightarrow\sqrt{x}-1\) min (vì 2020 > 0)

Lại có : \(\sqrt{x}-1\ge-1\forall x\)

Dấu "=" xảy ra <=> x = 0

Vậy Min\(\dfrac{P\left(x\right)}{2020\sqrt{x}}=\dfrac{-1}{2020}\Leftrightarrow x=0\)

2 tháng 8 2023

Ta có : \(\sqrt{x+1}\) có nghĩa khi `x >= -1`  Từ đk ta có :

\(x+2\left(1+\sqrt{x+1}\right)=x+1+2\sqrt{x+1}+1=\left(\sqrt{x+1}+1\right)^2\)

\(\Leftrightarrow\sqrt{x+2\left(1+\sqrt{x+1}\right)}=\sqrt{x+1}+1\)

\(x+2\left(1-\sqrt{x+1}\right)=x+1-2\sqrt{x+1}+1=\left(\sqrt{x+1}-1\right)^2\\ \Leftrightarrow\sqrt{x+2\left(1-\sqrt{x+1}\right)}=\left|\sqrt{x+1}-1\right|\)

Ta có : \(y=\sqrt{x+1}+1+\left|\sqrt{x+1}-1\right|\)  `(1)`

Ta bỏ dấu \(\left|\right|\) ở `1`

Ta có TH :

`-1<= x <= 0` ; lúc này \(\sqrt{x+1}-1\le0\)

nên : \(\left|\sqrt{x+1}-4\right|=1-\sqrt{x+1}\)

`1` trở thành : `y=2`

`x>0` lúc này \(\sqrt{x+1}-1>0\) nên

\(\left|\sqrt{x+1}-1\right|=\sqrt{x+1}-1\)

`1` trở thành : \(y=2\sqrt{x+1}>2\left(x>0\right)\)

Vì : \(y=\left\{{}\begin{matrix}2khi-1\le x\le0\\2\sqrt{x+1}kh\text{i}>0\end{matrix}\right.\)

gtnn của `y=2` với mọi \(x\in\left[-1;0\right]\)

NV
21 tháng 3 2022

ĐKXĐ: \(x\ge0;x\ne1\)

\(M=\left(\dfrac{1}{\sqrt{x}+1}-\dfrac{2\sqrt{x}-2}{\left(x-1\right)\left(\sqrt{x}+1\right)}\right):\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)

\(=\left(\dfrac{x-1-2\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(x-1\right)}\right):\left(\dfrac{\sqrt{x}+1-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)

\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}:\dfrac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2}.\left(\sqrt{x}+1\right)=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)

b.

\(M=\dfrac{\sqrt{x}+1-2}{\sqrt{x}+1}=1-\dfrac{2}{\sqrt{x}+1}\ge1-\dfrac{2}{0+1}=-1\)

\(M_{min}=-1\) khi \(x=0\)

I don't now

sorry

.....................

I don't know . Sorry !!!!

C=|x-2021|+|1-x|>=|x-2021+1-x|=2020

Dấu = xảy ra khi 1<=x<=2021

27 tháng 6 2023

bạn ơi, tìm gtnn của biểu thức ạ ;-;

\(B=\left(\frac{x\sqrt{x}+x+\sqrt{x}}{x\sqrt{x}-1}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\right).\frac{x-1}{2x+\sqrt{x}-1}\)  ĐKXĐ:...
Đọc tiếp

\(B=\left(\frac{x\sqrt{x}+x+\sqrt{x}}{x\sqrt{x}-1}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\right).\frac{x-1}{2x+\sqrt{x}-1}\)  ĐKXĐ: ...

\(=\frac{\left(x\sqrt{x}+x+\sqrt{x}\right)\left(1-\sqrt{x}\right)-\left(\sqrt{x}+3\right)\left(x\sqrt{x}-1\right)}{\left(x\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}.\frac{x-1}{2x+2\sqrt{x}-\sqrt{x}-1}\)

\(=\frac{x\sqrt{x}+x+\sqrt{x}-x^2-x\sqrt{x}-x-x^2+\sqrt{x}-3x\sqrt{x}+3}{\left(x\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}.\frac{x-1}{2\sqrt{x}\left(\sqrt{x}+1\right)-\left(\sqrt{x}+1\right)}\)

\(=\frac{-3x\sqrt{x}+2\sqrt{x}-2x^2+3}{\left(x\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}.\frac{x-1}{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{3-3x\sqrt{x}+2\sqrt{x}-2x^2}{\left(x\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}.\frac{1}{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{3\left(1-x\sqrt{x}\right)+2\sqrt{x}\left(1-x\sqrt{x}\right)}{\left(x\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}.\frac{1}{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{\left(2\sqrt{x}+3\right)\left(1-x\sqrt{x}\right)}{\left(x\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}.\frac{x-1}{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{-2\sqrt{x}-3}{1-\sqrt{x}}.\frac{x-1}{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{-2\sqrt{x}-3}{1-\sqrt{x}}.\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2\sqrt{x}-1}\)

\(=\frac{2\sqrt{x}+3}{2\sqrt{x}-1}\)

1
23 tháng 5 2019

hỏi j v

1:

a: \(A=\dfrac{\sqrt{x}+1-2}{\sqrt{x}+1}=1-\dfrac{2}{\sqrt{x}+1}\)

căn x+1>=1

=>2/căn x+1<=2

=>-2/căn x+1>=-2

=>A>=-2+1=-1

Dấu = xảy ra khi x=0

b: loading...