a/-x⁴+x³-16x+1

b/-77

c/\(\frac{-4x^2+3}{4}\)

 b,\(\Rightarrow\)\(\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}\right):2=\frac{2013}{2015}:2\)

\(\Rightarrow\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}=\frac{2013}{4030}\)

\(\Rightarrow\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x.\left(x+1\right)}=\frac{2013}{4030}\)

\(\Rightarrow\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{2013}{4030}\)

\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{2013}{4030}\)

\(\Rightarrow\frac{1}{x+1}=\frac{1}{2015}\)

\(\Rightarrow\)\(x+1=2015\)

\(\Rightarrow x=2014\)

a, 2/3x -3/2.x-1/2x=5/12

    x.(2/3-3/2-1/2)=5/12

                 x. -4/3=5/12

                          x=5/12:-4/3

                          x=-5/16

b,2/6+2/12+2/20+...+2/x.(x+1)=2013/2015

   2/2.3+2/3.4+2/4.5+...+2/x.(x+1)=2013/2015

   1/2(1-1/3+1/3-1/4+1/4-1/5+...+1/x-1/x+1)=2013/2015

                                                1/2(1-1/x+1)=2013/2015

                                                 1-1/x+1=2013/2015 : 1/2

                                                  1-1/x+1=4206/2015

                                                      suy ra đề sai

a) \(x^3-\frac{4}{25}x=0\)

\(\Leftrightarrow x\left(x+\frac{2}{5}\right)\left(x-\frac{2}{5}\right)=0\)

<=> x = 0

Xét 2 trường hợp: 

\(\Leftrightarrow x+\frac{2}{5}=0\)

      \(x=0-\frac{2}{5}\)

      \(x=-\frac{2}{5}\)

\(\Leftrightarrow x-\frac{2}{5}=0\)

      \(x=0+\frac{2}{5}\)

      \(x=\frac{2}{5}\)

\(\Rightarrow\orbr{\begin{cases}x=0\\x=\pm\frac{2}{5}\end{cases}}\)

b) \(\left(\frac{3}{8}+\frac{-3}{4}+\frac{7}{12}\right):\frac{5}{6}+\frac{1}{2}\)

\(=\left(\frac{3}{8}+\frac{-3}{4}+\frac{7}{12}\right):\frac{4}{3}\)

\(=\frac{13}{40}:\frac{4}{3}\)

\(=\frac{39}{120}=\frac{13}{40}\)

c) \(4\left(\frac{-1}{2}\right)^3-2\left(\frac{-1}{2}\right)^2+3\left(\frac{-1}{2}\right)-1\left(\frac{-1}{2}\right)^0\)

\(=4\left(\frac{-1}{2}\right)^3-2\left(\frac{-1}{2}\right)^3+3\left(\frac{-1}{2}\right)-1.1\)

\(=-\frac{1}{2}-\frac{1}{2}-\frac{3}{2}-1.1\)

\(=-\frac{5}{2}-1\)

\(=-\frac{7}{2}\)

\(x^3+3x^2+3x+1+y^3+3y^3+3y+1+x+y+2=0\)

\(\Leftrightarrow\left(x+1\right)^3+\left(y+1\right)^3+x+y+2=0\)

\(\Leftrightarrow\left(x+y+2\right)\left(\left(x+1\right)^2+\left(y+1\right)^2-\left(x+1\right)\left(y+1\right)\right)+\left(x+y+2\right)=0\)

\(\Leftrightarrow\left(x+y+2\right)\left(\left(x+1\right)^2+\left(y+1\right)^2-\left(x+1\right)\left(y+1\right)+1\right)=0\)

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

(phần trong ngoặc \(\left(x+1\right)^2-\left(x+1\right)\left(y+1\right)+\frac{\left(y+1\right)^2}{4}+\frac{3\left(y+1\right)^2}{4}+1\)

\(=\left(x+1-\frac{y+1}{4}\right)^2+\frac{3\left(y+1\right)^2}{4}+1\) luôn dương)

\(\Rightarrow x+y=-2\)

Mà \(xy>0\Rightarrow\left\{{}\begin{matrix}x< 0\\y< 0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}-x>0\\-y>0\end{matrix}\right.\)

Ta có: \(\frac{1}{-x}+\frac{1}{-y}\ge\frac{4}{-\left(x+y\right)}=2\) \(\Leftrightarrow\frac{1}{x}+\frac{1}{y}\le-2\) (đpcm)

Dấu "=" xảy ra khi và chỉ khi \(x=y=-1\)

2/ \(x;y;z\ne0\)

\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\)

\(\Leftrightarrow\frac{x+y}{xy}+\frac{1}{z}-\frac{1}{x+y+z}=0\)

\(\Leftrightarrow\frac{x+y}{xy}+\frac{x+y}{xz+yz+z^2}=0\)

\(\Leftrightarrow\left(x+y\right)\left(\frac{1}{xy}+\frac{1}{xz+yz+z^2}\right)=0\)

\(\Leftrightarrow\left(x+y\right)\left(\frac{xy+yz+xz+z^2}{xyz\left(x+y+z\right)}\right)=0\)

\(\Leftrightarrow\frac{\left(x+y\right)\left(y+z\right)\left(z+x\right)}{xyz\left(x+y+z\right)}=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-y\\y=-z\\z=-x\end{matrix}\right.\) dù trường hợp nào thì thay vào ta đều có \(B=0\)

3/ \(\Leftrightarrow mx-2x+my-y-1=0\)

\(\Leftrightarrow m\left(x+y\right)-\left(2x+y+1\right)=0\)

Gọi \(A\left(x_0;y_0\right)\) là điểm cố định mà d đi qua

\(\Leftrightarrow\left\{{}\begin{matrix}x_0+y_0=0\\2x_0+y_0+1=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x_0=-1\\y_0=1\end{matrix}\right.\)

Vậy d luôn đi qua \(A\left(-1;1\right)\) với mọi m

A=\(\left(1+x\right)\left(1+\frac{1}{y}\right)+\left(1+\frac{1}{x}\right)\left(1+y\right)=x+\frac{x}{y}+\frac{1}{y}+1+y+\frac{y}{x}+\frac{1}{x}+1\)

=\(\left(x+y+\frac{1}{x}+\frac{1}{y}\right)+\frac{x}{y}+\frac{y}{x}+2\)

mà x²+y²=1

=>2(x²+y²)>(=)(x+y)²

\(\Rightarrow x+y\le\sqrt{2}\)

áp dụng bất đẳng thức cô si ta có:

\(\left(x+y+\frac{1}{x}+\frac{1}{y}\right)+\frac{x}{y}+\frac{y}{x}+2\ge\left(x+y+\frac{4}{x+y}\right)+4\)

                                                                            \(=\left[\left(x+y\right)+\frac{2}{x+y}+\frac{2}{x+y}\right]+4\ge2\sqrt{2}+\sqrt{2}+4=4+3\sqrt{2}\)

Câu hỏi của Nguyễn Quỳnh Nga - Toán lớp 9 - Học toán với OnlineMath

Giải:

Ta có:

\(S=2+x+y+\frac{1}{x}+\frac{1}{y}+\frac{x}{y}+\frac{y}{x}\ge4+x+y+\frac{1}{x}+\frac{1}{y}\)

Mặt khác ta có: \(x+\frac{1}{2x}\ge\sqrt{2}\)

                        \(y+\frac{1}{2y}\ge\sqrt{2}\)

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

Cộng vế theo vế ta có ĐPCM

Dấu "=" xảy ra \(\Leftrightarrow x=y=\frac{\sqrt{2}}{2}\)