\(\frac{1}{a}=x;\frac{1}{b}=y;\frac{1}{c}=z\)

Viết lại đề như sau: \(\hept{\begin{cases}x+y+z=3\\2xy-z^2=9\end{cases}}\)

\(\Leftrightarrow x^2+y^2+z^2+2xy+2yz+2xz-2xy+z^2=0\)

\(\Leftrightarrow x^2+y^2+2z^2+2yz+2xz=0\)

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

\(\Leftrightarrow x=y=-z\Leftrightarrow\frac{1}{a}=\frac{1}{b}=-\frac{1}{c}\)

\(\Leftrightarrow a=b=-c\)

\(M=\left(a-3b+c\right)^{2018}=\left(a-3a-a\right)^{2018}=\left(3a\right)^{2018}\)

a, PTHH:\(Cu+Cl_2\rightarrow CuCl_2\)

b, \(n_{Cu}=\frac{m}{M}=\frac{12,8}{64}=0,2\left(mol\right)\)

Ta thấy \(n_{CuCl_2}=n_{Cl_2}=n_{Cu}=0,2\left(mol\right)\)

\(V_{Cl_2}=n_{Cl_2}.22,4=0,2.22,4=4,48\left(l\right)\)

c, \(m_{CuCl_2}=n_{CuCl_2}.M=0,2.135=27\left(g\right)\)

ta có:

\(\frac{b^2-c^2}{\left(a+b\right).\left(a+c\right)}=\frac{b^2-a^2+a^2-c^2}{\left(a+b\right).\left(a+c\right)}=\frac{\left(b-a\right).\left(b+a\right)+\left(a-c\right).\left(a+c\right)}{\left(a+b\right).\left(a+c\right)}=\frac{b-a}{a+c}+\frac{a-c}{a+b}\left(1\right)\)

\(\frac{c^2-a^2}{\left(b+c\right).\left(b+a\right)}=\frac{c^2-b^2+b^2-a^2}{\left(b+c\right).\left(b+a\right)}=\frac{\left(c-b\right).\left(b+c\right)+\left(b-a\right).\left(a+b\right)}{\left(b+c\right).\left(b+a\right)}=\frac{c-b}{b+a}+\frac{b-a}{b+c}\left(2\right)\)

\(\frac{a^2-b^2}{\left(c+a\right).\left(c+b\right)}=\frac{a^2-c^2+c^2-b^2}{\left(c+a\right).\left(c+b\right)}=\frac{\left(a-c\right).\left(a+c\right)+\left(c-b\right).\left(c+b\right)}{\left(c+a\right).\left(c+b\right)}=\frac{a-c}{c+b}+\frac{c-b}{c+a}\left(3\right)\)

từ (1),(2),(3)

\(\Rightarrow\frac{b^2-c^2}{\left(a+b\right).\left(a+c\right)}+\frac{c^2-a^2}{\left(b+c\right).\left(b+a\right)}+\frac{a^2-b^2}{\left(c+a\right).\left(c+b\right)}\)

\(=\frac{b-a}{a+c}+\frac{a-c}{a+b}+\frac{c-b}{a+b}+\frac{b-a}{b+c}+\frac{a-c}{c+b}+\frac{c-b}{c+a}=\frac{c-a}{a+c}+\frac{b-c}{b+c}+\frac{a-b}{a+b}\Rightarrowđpcm\)

Ta có :

\(\frac{1}{2x^2-5x+4}=\frac{1}{2\left(x-\frac{5}{4}\right)^2+\frac{7}{8}}\)

Để \(\frac{1}{2x^2-5x+4}\)MAX thì \(2\left(x-\frac{5}{4}\right)^2+\frac{7}{8}\)đạt MIN

Mà \(2\left(x-\frac{5}{4}\right)^2+\frac{7}{8}\ge\frac{7}{8}\)( Do \(2\left(x-\frac{5}{4}\right)^2\ge0\Rightarrow2\left(x-\frac{5}{4}\right)^2+\frac{7}{8}\ge\frac{7}{8}\))

Dấu " = " xảy ra \(\Leftrightarrow x-\frac{5}{4}=0\Leftrightarrow x=\frac{5}{4}\)

=> \(\frac{1}{2x^2-5x+4}\le\frac{1}{\frac{7}{8}}=\frac{8}{7}\)

Vậy \(\frac{1}{2x^2-5x+4}\)MAX khi x = \(\frac{5}{4}\)

\(\frac{1}{2x^2-5x+4}\)

ta có: \(2x^2-5x+4=2.\left(x^2-\frac{5x}{2}\right)+4=2.\left(x^2-2.x.\frac{5}{4}+\frac{25}{16}\right)+\frac{7}{8}=2.\left(x-\frac{5}{4}\right)^2+\frac{7}{8}\ge\frac{7}{8}\)

\(\left(\frac{1}{2x^2-5x+4}\right)max\Rightarrow\left(2x^2-5x+4\right)min\)

Vì tử thức=1>0 và không đổi =>  \(\left(2x^2-5x+4\right)>0\)

mà \(\left(2x^2-5x+4\right)\)\(\ge\) \(\frac{7}{8}\)

dấu = xảy ra khi \(x-\frac{5}{4}=0\)

\(\Rightarrow x=\frac{5}{4}\)

Vậy Max\(\frac{1}{2x^2-5x+4}=\frac{1}{\frac{7}{8}}=\frac{8}{7}\)khi \(x=\frac{5}{4}\)

p/s: bn An hòa thiếu vài chỗ nhé =))

\(\left(xy+\frac{1}{xy}\right)^2-\left(x+\frac{1}{x}\right)\left(y+\frac{1}{y}\right)\left(xy+\frac{1}{xy}\right)\)

\(=\left(xy+\frac{1}{xy}\right)\left[\left(xy+\frac{1}{xy}\right)-\left(x+\frac{1}{x}\right)\left(y+\frac{1}{y}\right)\right]\)

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

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

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

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

\(=4\)

Vậy giá trị bt ko phụ thuộc vào biến

bn có thể giải thích rõ hơn tại sao lại bằng 4 được không? Dù gì thì cx cảm ơn bn đã tl câu hỏi của mk

Gọi M là giao điểm của DF và BC

\(\Delta BKC\)có BF là đường cao đồng thời là phân giác nên \(\Delta BKC\)cân tại B

\(\Rightarrow\)BF cũng là trung tuyến\(\Rightarrow KF=CF\)

Lại có AD = CD (gt) nên FD là đường trung bình của \(\Delta AKC\)

\(\Rightarrow FD//AK\)hay \(DF//KB\)và 2FD = AK

\(\Rightarrow\frac{BG}{DG}=\frac{BK}{FD}=\frac{2BK}{AK}\)(1)

Ta có: \(\frac{EC}{ED}=\frac{DC-DE}{DE}=\frac{DC}{DE}-1=\frac{AD}{DE}-1\)

\(=\frac{AE-DE}{DE}-1=\frac{AE}{DE}-2\)

DM // AB (cmt) \(\Rightarrow\frac{AE}{DE}=\frac{AB}{DF}\)

\(\Rightarrow\frac{AE}{DE}-2=\frac{AB}{DF}-2=\frac{AK+KB}{DF}-2\)

\(=\frac{2\left(AK+KB\right)}{AK}-2=2+\frac{2BK}{AK}-2=\frac{2BK}{AK}\)(2)

Từ (1) và (2) suy ra \(\frac{BG}{DG}=\frac{CE}{DE}\)

\(\Rightarrow GE//BC\)(theo định lý Thales đảo)

Vậy \(GE//BC\)(đpcm)