G = \(\dfrac{x^2}{x-1}\)

= \(\dfrac{x^2-4x+4+4x-4}{x-1}\)

= \(\dfrac{\left(x-2\right)^2+4\left(x-1\right)}{x-1}\)

= \(\dfrac{\left(x-2\right)^2}{x-1}+4\)

Vì x>1 nên \(\left\{{}\begin{matrix}\left(x-2\right)^2\text{≥}0\\x-1>0\end{matrix}\right.\)

=> G ≥ 4

=> G = 4 đạt GTNN

Dấu bằng xảy ra <=> \(\left(x-2\right)^2=0\)

<=> \(x=2\)

\(Do\) \(x>2\)

\(=>\left\{{}\begin{matrix}x-2\text{ ≥0}\\2x-1>0\end{matrix}\right.\)

\(=>\left(x-2\right)\left(2x-1\right)\text{ ≥0}\)

\(< =>2x^2-5x+2\text{≥}0\)

\(< =>2x^2+2\text{≥}5x\)

\(< =>2x+\dfrac{2}{x}\text{≥}5\)

\(< =>x+\dfrac{1}{x}\text{≥}2,5\)

\(< =>H\text{≥}2,5\)

\(< =>H=5\) \(đạt\) \(GTNN\)

Dấu bằng xảy ra khi \(x-2=0< =>x=2\)

\(K=x^2+\dfrac{1}{x}\)

\(=\dfrac{53x^3}{54}+\left(\dfrac{x^2}{54}+\dfrac{1}{2x}+\dfrac{1}{2x}\right)\)

Áp dụng BĐT Cô si cho 3 số dương

\(\dfrac{x^2}{54}+\dfrac{1}{2x}+\dfrac{1}{2x}\text{≥}3.\sqrt[3]{\dfrac{x^2}{54}.\dfrac{1}{2x}.\dfrac{1}{2x}}\)\(\text{≥}\dfrac{53.9}{54}+3.\sqrt[3]{54.4}\)\(=\dfrac{28}{3}\)

Dấu bằng xảy ra khi \(\left\{{}\begin{matrix}\dfrac{x^2}{54}=\dfrac{1}{2x}=\dfrac{1}{2x}\\x=3\end{matrix}\right.\)\(< =>x=3\)

Em gửi ảnh ạ !

Em gửi ảnh trên ạ !!!!!

a, Ta có \(x=49\Rightarrow\sqrt{x}=7\)

Thay vào biểu thức A ta được : 

\(A=\frac{7.4}{7-1}=\frac{28}{6}=\frac{14}{3}\)

b, Với \(x\ge0;x\ne1\)

\(B=\frac{1}{\sqrt{x}+1}+\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{2}{x-1}=\frac{\sqrt{x}-1+x+\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{x+2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)( đpcm )

1) Khi x = 49 thì:

\(A=\frac{4\sqrt{49}}{\sqrt{49}-1}=\frac{4\cdot7}{7-1}=\frac{28}{6}=\frac{14}{3}\)

2) Ta có:

\(B=\frac{1}{\sqrt{x}+1}+\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{2}{x-1}\)

\(B=\frac{\sqrt{x}-1+x+\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(B=\frac{x+2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(B=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)

c) \(P=A\div B=\frac{4\sqrt{x}}{\sqrt{x}-1}\div\frac{\sqrt{x}+1}{\sqrt{x}-1}=\frac{4\sqrt{x}}{\sqrt{x}+1}\)

Ta có: \(P\left(\sqrt{x}+1\right)=x+4+\sqrt{x-4}\)

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

\(\Leftrightarrow4\sqrt{x}=x+4+\sqrt{x-4}\)

\(\Leftrightarrow\left(\sqrt{x}-2\right)^2+\sqrt{x-4}=0\)

Mà \(VT\ge0\left(\forall x\ge0,x\ne1\right)\)

\(\Rightarrow\hept{\begin{cases}\left(\sqrt{x}-2\right)^2=0\\\sqrt{x-4}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}\sqrt{x}=2\\x-4=0\end{cases}}\Rightarrow x=4\)

Vậy x = 4

Lời giải:

Áp dụng bổ đề sau:

Cho $a,b\geq 1$. Khi đó ta có $\frac{1}{a^2+1}+\frac{1}{b^2+1}\geq \frac{2}{ab+1}$

Bổ đề này có thể CM dễ dàng bằng cách biến đổi tương đương.

----------------------------

Áp dụng bổ đề trên ta có:

\(\frac{1}{x^2+1}+\frac{1}{y^2+1}\geq \frac{2}{xy+1}\)

\(\frac{1}{z^3+1}+\frac{1}{xyz+1}\geq \frac{2}{z^2\sqrt{xy}+1}\geq \frac{2}{z^2xy+1}\)

\(\frac{2}{xy+1}+\frac{2}{z^2xy+1}\geq \frac{4}{xyz+1}\)

\(\Rightarrow \frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^3+1}+\frac{1}{xyz+1}\geq \frac{4}{xyz+1}\)

\(\Rightarrow \frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^3+1}\geq \frac{3}{xyz+1}\) (đpcm)

Vậy.........

Bạn lưu ý lần sau viết đề bằng công thức toán để được hỗ trợ tốt hơn. Nhìn những đề viết kiểu này làm rất nản!

a) \(\sqrt{9a^4}=\sqrt{\left(3a^2\right)^2}=\left|3a^2\right|=3a^2\)

b) \(2\sqrt{a^2}-5a=2\left|a\right|-5a=-2a-5a=-7a\)

c) \(\sqrt{16\left(1+4x+4x^2\right)}=\sqrt{\left[4\left(1+2x\right)\right]^2}=\left|4\left(1+2x\right)\right|=4\left(1+2x\right)\)

a) \(M=\sqrt{4\left(x-1\right)}-\sqrt{9\left(x-1\right)}-\sqrt{16\left(x-1\right)}\)

\(=2\sqrt{x-1}-3\sqrt{x-1}-4\sqrt{x-1}=-5\sqrt{x-1}\)

b) \(N=\sqrt{25\left(y+4\right)}+\sqrt{36\left(y+4\right)}-2\sqrt{81\left(y+4\right)}\)

\(=5\sqrt{y+4}+6\sqrt{y+4}-18\sqrt{y+4}=-7\sqrt{y+4}\)

c) \(P=\sqrt{y-2}-3\sqrt{64\left(y-2\right)}+4\sqrt{49\left(y-2\right)}\)

\(=\sqrt{y-2}-24\sqrt{y-2}+28\sqrt{y-2}=5\sqrt{y-2}\)

a) \(M=\sqrt{4\left(x-1\right)}-\sqrt{9\left(x-1\right)}-\sqrt{16\left(x-1\right)}.\)

\(M=\sqrt{4\left(x-1\right)}-\sqrt{9\left(x-1\right)}-\sqrt{16\left(x-1\right)}\)

\(=2\sqrt{x-1}-3\sqrt{x-1}-4\sqrt{x-1}\)

\(=-5\sqrt{x-1}\)

b) \(N=\sqrt{25\left(y+4\right)}+\sqrt{36\left(y+4\right)}-2\sqrt{81\left(y+4\right)}\)

\(N=\sqrt{25\left(y+4\right)}+\sqrt{36\left(y+4\right)}-2\sqrt{81\left(y+4\right)}\)

\(=5\sqrt{y+4}+6\sqrt{y+4}\)

\(=-7\sqrt{y+4}\)

c) \(P=\sqrt{\left(y-2\right)}-3\sqrt{64\left(y-2\right)}+4\sqrt{49\left(y-2\right)}\)

\(P=\sqrt{\left(y-2\right)}-3\sqrt{64\left(y-2\right)}+4\sqrt{49\left(y-2\right)}\)

\(=\sqrt{y-2}-24\sqrt{y-2}+28\sqrt{y-2}\)

\(=5\sqrt{y-2}\)

M=-5√(x-1)

N=-7√(y+4)

P=-2√(y-2)

sử dụng phương pháp miền giá trị

bạn nói rõ hơn được không?

\(3a.P=\left(\frac{x-6}{x+3\sqrt{x}}-\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x}+3}\right):\frac{2\sqrt{x}-6}{x+1}\\ =\left(\frac{3\left(x-6\right)}{3\left(x+3\sqrt{x}\right)}-\frac{1\left(3\sqrt{x}+9\right)}{\sqrt{x}\left(3\sqrt{x}+9\right)}+\frac{1.3\sqrt{x}}{3\sqrt{x}\left(\sqrt{x}+3\right)}\right):\frac{2\sqrt{x}-6}{x+1}\\ =\left(\frac{3x-18}{3x+9\sqrt{x}}-\frac{3\sqrt{x}+9}{3x+9\sqrt{x}}+\frac{3\sqrt{x}}{3x+9\sqrt{x}}\right):\frac{2\sqrt{x}-6}{x+1}\\ =\left(\frac{3x-18-3\sqrt{x}-9+3\sqrt{x}}{3x+9\sqrt{x}}\right):\frac{2\sqrt{x}-6}{x+1}\\ =\frac{3x-27}{3x+9\sqrt{x}}:\frac{2\sqrt{x}-6}{x+1}\\ =\frac{3\left(x-9\right)}{3\left(x+3\sqrt{x}\right)}:\frac{2\sqrt{x}-6}{x+1}\\ \)

\(=\frac{x-9}{x+3\sqrt{x}}:\frac{2\sqrt{x}-6}{x+1}\\ =\frac{x-9}{x+3\sqrt{x}}.\frac{x+1}{2\sqrt{x}-6}\\ =\frac{\left(x-9\right)\left(x+1\right)}{\left(x+3\sqrt{x}\right)\left(2\sqrt{x}-6\right)}\\ =\frac{\left(x-9\right)\left(x+1\right)}{2x\sqrt{x}-18\sqrt{x}}\\ =\frac{\left(x-9\right)\left(x+1\right)}{2\sqrt{x}\left(x-9\right)}\\ =\frac{x+1}{2\sqrt{x}}\)

b) P=1

\(\frac{x+1}{2\sqrt{x}}=1\\ \Leftrightarrow2\sqrt{x}=x+1\\ \Leftrightarrow4x=x^2+2x+1\\ \Leftrightarrow-x^2+2x-1=0\\ \Leftrightarrow x^2-2x+1=0\\ \Leftrightarrow\left(x-1\right)^2=0\\ \Leftrightarrow x-1=0\\ \Leftrightarrow x=1\)

Kết luận: P=1 thì x nhận giá trị.....

Chúc bạn học tốt , thắc mắc hỏi bên dưới nhé !