\(N=\frac{a}{1+b^2c}+\frac{b}{1+c^2d}+\frac{c}{1+d^2a}+\frac{d}{1+a^2b}\)

Áp dụng BĐT Cauchy ta có:

\(\frac{a}{1+b^2c}=a-\frac{ab^2c}{1+b^2c}\)

\(\ge a-\frac{ab^2c}{2b\sqrt{c}}=a-\frac{ab\sqrt{c}}{2}=a-\frac{b\sqrt{ac}\sqrt{a}}{2}\)

\(\ge a-\frac{b\left(ac+c\right)}{4}\).Suy ra \(\frac{a}{1+b^2c}\ge a-\frac{1}{4}\cdot\left(ab+abc\right)\)

Tương tự ta có:

\(\frac{b}{a+c^2d}\ge b-\frac{1}{4}\left(bc+bcd\right)\)

\(\frac{c}{1+d^2a}\ge c-\frac{1}{4}\left(cd+cda\right)\)

\(\frac{d}{1+a^2b}\ge d-\frac{1}{4}\left(da+dab\right)\)

Do đó: \(S=\frac{a}{1+b^2c}+\frac{b}{1+c^2d}+\frac{c}{1+d^2a}+\frac{d}{1+a^2b}\)

\(\ge a+b+c+d-\frac{1}{4}\left(ab+bc+cd+da+abc+bcd+cda+dab\right)\)

\(=4-\frac{1}{4}\left(ab+bc+cd+da+abc+bcd+cda+dab\right)\)

Ta có:

\(ab+bc+cd+da\le\frac{1}{4}\left(a+b+c+d\right)^2=4\)

\(abc+bcd+cda+dab\le\frac{1}{16}\left(a+b+c+d\right)^3=4\)

nên \(S\ge4-\frac{1}{4}\cdot\left(4+4\right)=2\)(Đpcm)

Dấu = khi \(a=b=c=d=1\)

tick đê =))

Cách 1. Áp dụng bđt Bunhiacopxki : \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge\left(\sqrt{a.\frac{1}{a}}+\sqrt{b.\frac{1}{b}}+\sqrt{c.\frac{1}{c}}\right)^2=\left(1+1+1\right)^2=9\)

Cách 2. Áp dụng bđt Cauchy : 

\(a+b+c\ge3\sqrt[3]{abc}\)

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{3}{\sqrt[3]{abc}}\)

\(\Rightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\)

Bđt cauchy đi

ĐKXĐ : \(x,y>0\)

a/ \(A=\left(\sqrt{x}+\frac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-x}+\frac{x+y}{\sqrt{xy}}\right)\)

\(=\left(\frac{x+\sqrt{xy}+y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right).\sqrt{x}}-\frac{y\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}.\sqrt{y}\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}-\frac{\left(x+y\right)\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\right)\)

\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\frac{x^2-x\sqrt{xy}-y\sqrt{xy}-y^2-x^2+y^2}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\frac{-\sqrt{xy}\left(x+y\right)}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}.\frac{-\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{x+y}=\sqrt{y}-\sqrt{x}\)

b/ Ta có ; \(4+2\sqrt{3}=\left(\sqrt{3}+1\right)^2\)

\(\Rightarrow B=\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{3}=\sqrt{3}+1-\sqrt{3}=1\)

a) \(A=\left(\frac{1}{\sqrt{a}-1}-\frac{1}{\sqrt{a}}\right):\left(\frac{\sqrt{a}+1}{\sqrt{a}-2}-\frac{\sqrt{a}+2}{\sqrt{a}-1}\right)\left(ĐK:a>0;a\ne1;a\ne4\right)\)

\(=\frac{\sqrt{a}-\sqrt{a}+1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)-\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}\)

\(=\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{a-1-a+4}{\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}\)

\(=\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\frac{\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}{3}=\frac{\sqrt{a}-2}{3\sqrt{a}}\)

b) Để \(A>\frac{1}{6}\)

\(\Leftrightarrow\)\(\frac{\sqrt{a}-2}{3\sqrt{a}}>\frac{1}{6}\)

\(\Leftrightarrow\)\(\frac{\sqrt{a}-2}{3\sqrt{a}}-\frac{1}{6}>0\)

\(\Leftrightarrow\)\(\frac{2\sqrt{a}-4-\sqrt{a}}{6\sqrt{a}}>0\)

\(\Leftrightarrow\)\(\frac{\sqrt{a}-4}{6\sqrt{a}}>0\)

\(\Leftrightarrow\sqrt{a}-4>0\Leftrightarrow a>16\left(tm\right)\)

Vậy a>16 thì \(A>\frac{1}{6}\)

ĐKXĐ : \(a>0,a\ne4,a\ne1\)

a) \(A=\left(\frac{1}{\sqrt{a}-1}-\frac{1}{\sqrt{a}}\right):\left(\frac{\sqrt{a}+1}{\sqrt{a}-2}-\frac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)

\(=\frac{\sqrt{a}-\sqrt{a}+1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\left(\frac{a-1-\left(a-4\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}\right)\)

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

b) \(A>\frac{1}{6}\Rightarrow\frac{\sqrt{a}-2}{3\sqrt{a}}>\frac{1}{6}\Rightarrow-\frac{2}{3\sqrt{a}}+\frac{1}{3}>\frac{1}{6}\Rightarrow\frac{2}{3\sqrt{a}}>\frac{1}{6}\Rightarrow\frac{1}{\sqrt{a}}>\frac{1}{4}\Rightarrow a< 16\)

Kết hợp với điều kiện xác định.

Từ B kẻ BD vuông góc với BD , cắt CA tại D. 

=> Tam giác BCD vuông tại B có đường trung tuyến AB

=> AB = AC = AD

Ta có : \(\begin{cases}AH\text{//}BD\\AC=AD\end{cases}\) => AH là đường trung bình của tam giác BCD

=> \(AH=\frac{1}{2}BD\Rightarrow AH^2=\frac{BD^2}{4}\Rightarrow BD^2=4AH^2\)

Áp dụng hệ thức về cạnh trong tam giác vuông BDC có : 

\(\frac{1}{BK^2}=\frac{1}{BC^2}+\frac{1}{BD^2}\Leftrightarrow\frac{1}{BK^2}=\frac{1}{BC^2}+\frac{1}{4AH^2}\) 

he thuc lg la ra ngay

a/ Đặt BH = x (x>0) (đvđd) => CH = 100-x (đvđd)

Áp dụng hệ thức về cạnh trong tam giác ta có : \(BH.HC=AH^2\) hay 

\(x\left(100-x\right)=48^2\Leftrightarrow x^2-100x+48^2=0\) \(\Leftrightarrow\left[\begin{array}{nghiempt}x=36\\x=64\end{array}\right.\)

1. Nếu x = 36 thì \(AB=\sqrt{AH^2+BH^2}=\sqrt{48^2+36^2}=60\)

\(AC=\sqrt{AH^2+CH^2}=\sqrt{48^2+64^2}=80\)

2. Nếu x = 64 thì AB = 80 , AC = 60

b/ Ta có : góc ABD = góc ACB => góc ABD + góc ABC = góc ACB + góc ABC = 90 độ

=> BC vuông góc với BD tại B

Áp dụng hệ thức về cạnh trong tam giác vuông BDC có đường cao AB : 

\(\frac{1}{AB^2}=\frac{1}{BD^2}+\frac{1}{BC^2}\)(đpcm)

a)Đặt \(A=\sqrt{x-2}+\sqrt{4-x}\)

Đk:\(2\le x\le4\)

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

\(=2+2\sqrt{\left(x-2\right)\left(4-x\right)}\) (dùng BĐT Cauchy)

\(\le2+\left(x-2\right)+\left(4-x\right)\)

\(=2+2=4\)

\(\Rightarrow A^2\le4\Leftrightarrow A\le2\)

Dấu = khi \(\sqrt{x-2}=\sqrt{4-x}\Leftrightarrow x=3\)

Vậy MaxA=2 khi x=3

b)Đặt \(B=\sqrt{6-x}+\sqrt{x+2}\)

Đk:\(-2\le x\le6\)

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

\(=8+2\sqrt{\left(6-x\right)\left(x+2\right)}\) (Bđt Cauchy)

\(\le8+\left(6-x\right)+\left(x+2\right)\)

\(=8+8=16\)

\(\Rightarrow B^2\le16\Leftrightarrow B\le4\)

Dấu = khi \(\sqrt{6-x}=\sqrt{x+2}\Leftrightarrow x=2\)

Vậy MaxB=4 khi x=2

c)Đặt \(C=\sqrt{x}+\sqrt{2-x}\)

Đk:\(0\le x\le2\)

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

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

\(\le2+x+\left(2-x\right)\)

\(=2+2=4\)

\(\Rightarrow C^2\le4\Leftrightarrow C\le2\)

Dấu = khi \(\sqrt{x}=\sqrt{2-x}\Leftrightarrow x=1\)

Vậy MaxC=2 khi x=1

Xét vế trái : \(\left(\sqrt{n+1}-\sqrt{n}\right)^2=2n+1-2\sqrt{n}.\sqrt{n+1}\)

Xét vế phải : \(\sqrt{\left(2n+1\right)^2}-\sqrt{\left(2n+1\right)^2-1}=\left|2n+1\right|-\sqrt{\left(2n+1-1\right)\left(2n+1+1\right)}=2n+1-2\sqrt{n}.\sqrt{n+1}\)

=> VT = VP 

=> đpcm

khó quá ms đầu năm s học cao thế bạn ơi 

Ai giải giúp mình với ạ

Đề có sai ko ?

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

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

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

\(\Leftrightarrow1+x^2+y^2+x^2y^2=1+x^2y^2-2xy\)

\(\Leftrightarrow x^2+y^2=-2xy\)

\(\Leftrightarrow x^2+y^2+2xy=0\)

\(\Leftrightarrow\left(x+y\right)^2=0\)

\(\Leftrightarrow x=-y\)

Thay vào ,ta có

\(x\sqrt{1+y^2}+y\sqrt{1+x^2}=-y\sqrt{1+x^2}+y\sqrt{1+x^2}=0\)(đpcm)

đây là cách của mk

@-@

Ta có \(1=\left(xy+\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\right)^2\)

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

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

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

\(\Leftrightarrow x^2\left(1+y^2\right)+y^2\left(1+x^2\right)+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}=0\)

\(\Leftrightarrow\left(x\sqrt{1+y^2}+y\sqrt{1+x^2}\right)^2=0\)

\(\Rightarrow x\sqrt{1+y^2}+y\sqrt{1+x^2}=0\)