Ta có:

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

\(=\frac{1}{a^2+b^2+c^2}+\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{a^2+b^2+c^2}+\frac{9}{ab+bc+ca}\)

\(=\frac{1}{a^2+b^2+c^2}+\frac{1}{ab+bc+ca}+\frac{1}{ab+bc+ca}+\frac{7}{ab+bc+ca}\ge\frac{9}{\left(a+b+c\right)^2}+\frac{7}{ab+bc+ca}\)

DO:

\(\frac{9}{\left(a+b+c\right)^2}+\frac{7}{ab+bc+ca}\ge9+\frac{7}{\frac{\left(a+b+c\right)^2}{3}}=9+21=30\)

\(\Rightarrow DPCM\)

Tích t vs ku

Gọi tử số của phân số đó là \(x\left(x\in Z\right)\)

mẫu số của phân số đó là:\(x+3\)(do tử kém mẫu 3 đơn vị)

Vì  thêm cả tử và mẫu của phân số đã cho 2 đơn vị thì ta được 1 phân số mới có giá trị bằng \(\frac{1}{2}\),nên ta có phương trình:

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

\(\Leftrightarrow\frac{x+2}{x+5}=\frac{1}{2}\)

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

\(\Leftrightarrow2x+4=x+5\)

\(\Leftrightarrow2x-x=5-4\)

\(\Leftrightarrow x=1\left(tm\right)\)

Vậy phân số đó là:\(\frac{1}{\left(1+3\right)}=\frac{1}{4}\)

Xong rồi đấy,bạn k cho mình nha

\(\frac{1+\frac{1}{x}}{1-\frac{1}{x}}=\left(1+\frac{1}{x}\right):\left(1-\frac{1}{x}\right)\)

\(=\frac{x+1}{x}:\frac{x-1}{x}=\frac{x+1}{x}.\frac{x}{x-1}=\frac{x+1}{x-1}\)

EZ, đề thanh hóa sáng nay ^^

Ta có: \(VT=\frac{1}{a^2+b^2+c^2}+\frac{a+b+c}{abc}\)

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

\(\ge\frac{1}{a^2+b^2+c^2}+\frac{9}{ab+bc+ca}\)

\(\Rightarrow VT\ge\frac{1}{a^2+b^2+c^2}+\frac{1}{ab+bc+ca}+\frac{1}{ab+bc+ca}+\frac{7}{ab+bc+ca}\)

\(\ge\frac{9}{\left(a+b+c\right)^2}+\frac{7.3}{\left(a+b+c\right)^2}=30\)

cách khác này

\(n^4+2n^3-13n^2-14n+24\)

\(=\left(n^4+2n^3-n^2-2n\right)-12n^2-12n+24\)

\(=\left(n-1\right)n\left(n+1\right)\left(n+2\right)-12n^2-12n+24⋮6\)

đề dài v~

1.

a) \(f\left(x\right)=5x^2-2x+1\)

\(5f\left(x\right)=25x^2-10x+5\)

\(5f\left(x\right)=\left(25x^2-10x+1\right)+4\)

\(5f\left(x\right)=\left(5x-1\right)^2+4\)

Mà  \(\left(5x-1\right)^2\ge0\)

\(\Rightarrow5f\left(x\right)\ge4\)

\(\Leftrightarrow f\left(x\right)\ge\frac{4}{5}\)

Dấu " = " xảy ra khi :

\(5x-1=0\Leftrightarrow x=\frac{1}{5}\)

Vậy ....

b)  \(P\left(x\right)=3x^2+x+7\)

\(3P\left(x\right)=9x^2+3x+21\)

\(3P\left(x\right)=\left(9x^2+3x+\frac{1}{4}\right)+\frac{83}{4}\)

\(3P\left(x\right)=\left(3x+\frac{1}{2}\right)^2+\frac{83}{4}\)

Mà  \(\left(3x+\frac{1}{2}\right)^2\ge0\)

\(\Rightarrow3P\left(x\right)\ge\frac{83}{4}\)

\(\Leftrightarrow P\left(x\right)\ge\frac{83}{12}\)

Dấu "=" xảy ra khi :

\(3x+\frac{1}{2}=0\Leftrightarrow x=-\frac{1}{6}\)

Vậy ...

c)  \(Q\left(x\right)=5x^2-3x-3\)

\(5Q\left(x\right)=25x^2-15x-15\)

\(\Leftrightarrow5Q\left(x\right)=\left(25x^2-15x+\frac{9}{4}\right)-\frac{69}{4}\)

\(\Leftrightarrow5Q\left(x\right)=\left(5x-\frac{3}{2}\right)^2-\frac{69}{4}\)

Mà  \(\left(5x-\frac{3}{2}\right)^2\ge0\)

\(\Rightarrow5Q\left(x\right)\ge\frac{-69}{4}\)

\(\Leftrightarrow Q\left(x\right)\ge-\frac{69}{20}\)

Dấu "=" xảy ra khi :

\(5x-\frac{3}{2}=0\Leftrightarrow x=0,3\)

Vậy ...

2.

a)  \(f\left(x\right)=-3x^2+x-2\)

\(-3f\left(x\right)=9x^2-3x+6\)

\(-3f\left(x\right)=\left(9x^2-3x+\frac{1}{4}\right)+\frac{23}{4}\)

\(-3f\left(x\right)=\left(3x-\frac{1}{2}\right)^2+\frac{23}{4}\)

Mà  \(\left(3x-\frac{1}{2}\right)^2\ge0\)

\(\Rightarrow-3f\left(x\right)\ge\frac{23}{4}\)

\(\Leftrightarrow f\left(x\right)\le\frac{23}{12}\)

Dấu "=" xảy ra khi :

\(3x-\frac{1}{2}=0\Leftrightarrow x=\frac{1}{6}\)

Vậy ...

b)  \(P\left(x\right)=-x^2-7x+1\)

\(-P\left(x\right)=x^2+7x-1\)

\(-P\left(x\right)=\left(x^2+7x+\frac{49}{4}\right)-\frac{53}{4}\)

\(-P\left(x\right)=\left(x+\frac{7}{2}\right)^2-\frac{53}{4}\)

Mà  \(\left(x+\frac{7}{2}\right)^2\ge0\)

\(\Rightarrow-P\left(x\right)\ge-\frac{53}{4}\)

\(\Leftrightarrow P\left(x\right)\le\frac{53}{4}\)

Dấu "=" xảy ra khi :

\(x+\frac{7}{2}=0\Leftrightarrow x=-\frac{7}{2}\)

Vậy ...

c)  \(Q\left(x\right)=-2x^2+x-8\)

\(-2Q\left(x\right)=4x^2-2x+16\)

\(-2Q\left(x\right)=\left(4x^2-2x+\frac{1}{4}\right)+\frac{63}{4}\)

\(-2Q\left(x\right)=\left(2x-\frac{1}{2}\right)^2+\frac{63}{4}\)

Mà :  \(\left(2x-\frac{1}{2}\right)^2\ge0\)

\(\Rightarrow-2Q\left(x\right)\ge\frac{63}{4}\)

\(\Leftrightarrow Q\left(x\right)\le-\frac{63}{8}\)

Dấu "=" xảy ra khi :

\(2x-\frac{1}{2}=0\Leftrightarrow x=\frac{1}{4}\)

Vậy ...