\(ĐK:x\ge0\)

\(y=x-4\sqrt{x}-1=\left(\sqrt{x}\right)^2-4\sqrt{x}+4-5=\left(\sqrt{x}-2\right)^2-5\ge-5\)

Đẳng thức xảy ra khi x = 4

ĐKXĐ : \(x\ge0\)

Ta có :

\(y=x-4\sqrt{x}-1\)

\(\Leftrightarrow y=x-2.2\sqrt{x}+4-5\)

\(\Leftrightarrow y=\left(\sqrt{x}-2\right)^2-5\ge-5\)

Dấu bằng xảy ra

\(\Leftrightarrow\sqrt{x}-2=0\)

\(\Leftrightarrow\sqrt{x}=2\)

\(\Leftrightarrow x=4\)

Vậy giá trị nhỏ nhất của biểu thức y = -5 \(\Leftrightarrow x=4\)

a) =2(a+b)

b) =2(a-2b)

c) =2(a+2b-3c)

d) =3(a-2b+3c)

e)= -4(a+2b+3c)

f) =-5(x+2xy+3y)

g) =-7(a+2ab+3b)

h) = 6(xy-2x-3y)

k) = 8(xy-3y+2x)

n)=9(ab-2a+1)

Bài 1:

a) Đặt \(6x+7=y\)

\(PT\Leftrightarrow y^2\left(y-1\right)\left(y+1\right)=72\)

\(\Leftrightarrow y^4-y^2-72=0\)

\(\Leftrightarrow\left(y^2-9\right)\left(y^2+8\right)=0\)

Mà \(y^2+8>0\left(\forall y\right)\)

\(\Rightarrow y^2-9=0\Leftrightarrow\left(y-3\right)\left(y+3\right)=0\Leftrightarrow\left(6x+4\right)\left(6x+10\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}6x+4=0\\6x+10=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=-\frac{2}{3}\\x=-\frac{5}{3}\end{cases}}\)

b) đk: \(x\ne\left\{-4;-5;-6;-7\right\}\)

\(PT\Leftrightarrow\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{1}{x+4}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Leftrightarrow\frac{1}{x+4}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Leftrightarrow\frac{3}{\left(x+4\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow x^2+11x+28=54\)

\(\Leftrightarrow x^2+11x-26=0\)

\(\Leftrightarrow\left(x+13\right)\left(x-2\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=-13\\x=2\end{cases}}\)

Bài 2 không tiện vẽ hình nên thôi nhờ godd khác:)

Bài 3:

Ta có:

\(a_n=1+2+3+...+n\)

\(a_{n+1}=1+2+3+...+n+\left(n+1\right)\)

\(\Rightarrow a_n+a_{n+1}=2\cdot\left(1+2+3+...+n\right)+\left(n+1\right)\)

\(=2\cdot\frac{n\left(n+1\right)}{2}+n+1\)

\(=n^2+n+n+1=\left(n+1\right)^2\)

Là SCP => đpcm