x+y+z=0

nên x+y=-z; y+z=-x; x+z=-y

\(\left(1+\dfrac{x}{y}\right)\left(1+\dfrac{y}{z}\right)\left(1+\dfrac{z}{x}\right)\)

\(=\dfrac{x+y}{y}\cdot\dfrac{y+z}{z}\cdot\dfrac{x+z}{x}=-1\)

Ta có: \(P=\dfrac{x}{x+1}+\dfrac{y}{y+1}+\dfrac{z}{z+1}\)

\(=\dfrac{x+1-1}{x+1}+\dfrac{y+1-1}{y+1}+\dfrac{z+1-1}{z+1}\)

\(=3-\left(\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}\right)\)

Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:

\(\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}\ge\dfrac{9}{x+y+z+3}=\dfrac{9}{4}\)

\(\Rightarrow P\le\dfrac{3}{4}\)

Đẳng thức xảy ra khi \(x=y=z=\dfrac{1}{3}\)

P/s: bài này có max ko có min vì khi cho hai trong ba số tiến gần đến không thì giá trị của biểu thức ngày càng nhỏ

ơ sao lại 3/4 hả bạn tưởng 9/4 chứ

nhanh giùm nhé!!!!!!!!!

duyệt đi

làm xong có thù lao ko =))

Chứng minh biểu thức thế nào em?

e vt thiếu , biểu thức có giá trị nguyên ạ

2: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)

=>\(\frac{xy+yz+xz}{xyz}=0\)

=>xy+yz+xz=0

=>xy=-xz-yz; yz=-xy-xz; xz=-xy-yz

\(x^2+2yz=x^2+yz+yz\)

\(=x^2+yz-xy-xz=x\left(x-y\right)-z\left(x-y\right)=\left(x-y\right)\left(x-z\right)\)

\(y^2+2xz=y^2+xz+xz\)

\(=y^2+xz-xy-yz=y^2-xy+xz-yz\)

=y(y-x)+z(x-y)

=z(x-y)-y(x-y)=(x-y)(z-y)

\(z^2+2xy\)

\(=z^2+xy+xy\)

\(=z^2+xy-yz-xz\)

\(=z^2-xz+xy-yz=z\left(z-x\right)+y\left(x-z\right)=\left(x-z\right)\left(y-z\right)\)

\(A=\frac{yz}{x^2+2yz}+\frac{xz}{y^2+2xz}+\frac{xy}{z^2+2xy}\)

\(=\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{xz}{\left(x-y\right)\left(z-y\right)}+\frac{xy}{\left(x-z\right)\left(y-z\right)}\)

\(=\frac{yz\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=\frac{y^2z-yz^2-x^2z+xz^2+x^2y-xy^2}{\left(x-y\right)\cdot\left(x-z\right)\left(y-z\right)}\)

\(=\frac{z\left(y^2-x^2\right)+z^2\left(x-y\right)+xy\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=\frac{\left(x-y\right)\left\lbrack-z\left(x+y\right)+z^2+xy\right\rbrack}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)

\(=\frac{-xz-yz+z^2+xy}{\left(x-z\right)\left(y-z\right)}=\frac{z^2-yz-xz+xy}{\left(x-z\right)\left(y-z\right)}=\frac{z\left(z-y\right)-x\left(z-y\right)}{\left(x-z\right)\left(y-z\right)}=\frac{\left(z-x\right)\left(z-y\right)}{\left(z-x\right)\left(z-y\right)}\)

=1

BĐT: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\)

Nếu ko bạn có thể làm theo AM-GM:

\(\frac{1}{1+x}+\frac{1+x}{4}\ge2\sqrt{\frac{1+x}{4\left(x+1\right)}}=1\)

Tương tự: \(\frac{1}{1+y}+\frac{1+y}{4}\ge1\) ; \(\frac{1}{1+z}+\frac{1+z}{4}\ge1\)

Cộng vế với vế:

\(A+\frac{3+x+y+z}{4}\ge3\Rightarrow A\ge3-\frac{3+x+y+z}{4}\ge3-\frac{3+3}{4}=\frac{3}{2}\)

Dấu "=" xảy ra khi \(x=y=z=1\)

\(A=\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge\frac{9}{1+x+1+y+1+z}=\frac{9}{3+x+y+z}\ge\frac{9}{3+3}=\frac{3}{2}\)

\(A_{min}=\frac{3}{2}\) khi \(x=y=z=1\)

Có \(x-y-z=0\)
\(\Rightarrow\left\{{}\begin{matrix}x-z=y\\x-y=z\\z+y=x\end{matrix}\right.\Rightarrow y-x=-z\)
Có x,y,z ≠ 0
\(\Rightarrow A=\left(1-\dfrac{z}{x}\right)\left(1-\dfrac{x}{y}\right)\left(1+\dfrac{y}{z}\right)\)
\(\Rightarrow A=\left(\dfrac{x-z}{x}\right)\left(\dfrac{y-x}{y}\right)\left(\dfrac{z+y}{z}\right)\)
\(\Rightarrow A=\left(\dfrac{y}{x}\right)\left(\dfrac{-z}{y}\right)\left(\dfrac{x}{z}\right)\)
\(\Rightarrow A=1\)
Vậy A = 1

A = -1
/ ấn nhầm ~ xin lỗi /
Vậy A = -1