em tham khảo

a: ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)

b: Ta có: \(A=\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}-1}{2}\)

\(=\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{2}{\sqrt{x}-1}\)

\(=\dfrac{2}{x+\sqrt{x}+1}\)

c: Ta có: \(x+\sqrt{x}+1>0\forall x\) thỏa mãn ĐKXĐ

\(\Leftrightarrow\dfrac{2}{x+\sqrt{x}+1}>0\forall x\)

1:

\(A=\dfrac{15\sqrt{x}-11-\left(3\sqrt{x}-2\right)\left(\sqrt{x}+3\right)-\left(2\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{15\sqrt{x}-11-3x-9\sqrt{x}+2\sqrt{x}+6-2x+2\sqrt{x}-3\sqrt{x}+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{-5x+7\sqrt{x}-2}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}=\dfrac{-5\sqrt{x}+2}{\sqrt{x}+3}\)

3: A nguyên

=>-5căn x-15+17 chia hết cho căn x+3

=>căn x+3 thuộc Ư(17)

=>căn x+3=17

=>x=196

\(a,Q=\dfrac{x-6\sqrt{x}+9+x+6\sqrt{x}+9+14}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\dfrac{\sqrt{x}-3}{x}\left(x>0;x\ne9\right)\\ Q=\dfrac{2x+32}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\dfrac{\sqrt{x}-3}{x}=\dfrac{2x+32}{x\left(\sqrt{x}+3\right)}\)

Câu 1:

Tìm max:

Áp dụng BĐT Bunhiacopxky ta có:

\(y^2=(3\sqrt{x-1}+4\sqrt{5-x})^2\leq (3^2+4^2)(x-1+5-x)\)

\(\Rightarrow y^2\leq 100\Rightarrow y\leq 10\)

Vậy \(y_{\max}=10\)

Dấu đẳng thức xảy ra khi \(\frac{\sqrt{x-1}}{3}=\frac{\sqrt{5-x}}{4}\Leftrightarrow x=\frac{61}{25}\)

Tìm min:

Ta có bổ đề sau: Với $a,b\geq 0$ thì \(\sqrt{a}+\sqrt{b}\geq \sqrt{a+b}\)

Chứng minh:

\(\sqrt{a}+\sqrt{b}\geq \sqrt{a+b}\)

\(\Leftrightarrow (\sqrt{a}+\sqrt{b})^2\geq a+b\)

\(\Leftrightarrow \sqrt{ab}\geq 0\) (luôn đúng).

Dấu "=" xảy ra khi $ab=0$

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

Áp dụng bổ đề trên vào bài toán ta có:

\(\sqrt{x-1}+\sqrt{5-x}\geq \sqrt{(x-1)+(5-x)}=2\)

\(\sqrt{5-x}\geq 0\)

\(\Rightarrow y=3(\sqrt{x-1}+\sqrt{5-x})+\sqrt{5-x}\geq 3.2+0=6\)

Vậy $y_{\min}=6$

Dấu "=" xảy ra khi \(\left\{\begin{matrix} (x-1)(5-x)=0\\ 5-x=0\end{matrix}\right.\Leftrightarrow x=5\)

Bài 2:

\(A=\sqrt{(x-1994)^2}+\sqrt{(x+1995)^2}=|x-1994|+|x+1995|\)

Áp dụng BĐT dạng \(|a|+|b|\geq |a+b|\) ta có:

\(A=|x-1994|+|x+1995|=|1994-x|+|x+1995|\geq |1994-x+x+1995|=3989\)

Vậy \(A_{\min}=3989\)

Đẳng thức xảy ra khi \((1994-x)(x+1995)\geq 0\Leftrightarrow -1995\leq x\leq 1994\)

Bài 3:

Ta thấy:

\(2x-x^2+7=8-(x^2-2x+1)=8-(x-1)^2\leq 8, \forall x\in\mathbb{R}\)

\(\Rightarrow 2+\sqrt{2x-x^2+7}\leq 2+\sqrt{8}=2+2\sqrt{2}\)

\(\Rightarrow B=\frac{3}{2+\sqrt{2x-x^2+7}}\geq \frac{3}{2+2\sqrt{2}}\)

Vậy GTNN của $B$ là \(\frac{3}{2+2\sqrt{2}}\).

Đẳng thức xảy ra tại \((x-1)^2=0\Leftrightarrow x=1\)

\(\dfrac{1}{M}=\dfrac{\sqrt{x}+2}{\sqrt{x}+5}\)

\(B=\dfrac{\sqrt{x}+2}{\sqrt{x}+5}-\dfrac{\sqrt{x}}{27}=\dfrac{27\sqrt{x}+54-x-5\sqrt{x}}{27\left(\sqrt{x}+5\right)}\)\(=\dfrac{-x+22\sqrt{x}+54}{27\left(\sqrt{x}+5\right)}\)

\(\Rightarrow\sqrt{x}.27B+135B=-x+22\sqrt{x}+54\)

\(\Leftrightarrow x+\sqrt{x}\left(27B-22\right)+135B-54=0\) (1)

Coi PT (1) là phương trình bậc 2 ẩn \(\sqrt{x}\)

PT (1) có nghiệm không âm \(\Leftrightarrow\left\{{}\begin{matrix}\Delta=729B^2-1728B+700\ge0\\S=22-27B\ge0\\P=135B-54\ge0\end{matrix}\right.\)

\(\Leftrightarrow\dfrac{2}{5}\le B\le\dfrac{14}{27}\)

Suy ra \(max_B=\dfrac{14}{27}\Leftrightarrow x=16\)

A làm tương tự 

ĐKXĐ:x\(\ge\)0

Ta có:\(\sqrt{x}\ge0\forall x\in R\)

=>-5\(\sqrt{x}\le0\forall x\in R\)

=>2-5\(\sqrt{x}\le2\forall x\in R\)

\(\sqrt{x}\ge0\forall x\in R\)

=>\(\sqrt{x}+3\ge3\forall x\in R\)

=>A\(=\dfrac{2-5\sqrt{x}}{\sqrt{x}+3}\le\dfrac{2}{3}\)

=>GTLN của A bằng \(\dfrac{2}{3}\) xảy ra khi và chỉ khi \(\sqrt{x}=0\)<=>x=0

Vậy...

a) Để biểu thức M có nghĩa thì \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)

b) \(M=\frac{2}{\sqrt{x}-1}+\frac{2\left(\sqrt{x}+1\right)}{x+\sqrt{x}+1}+\frac{x-10\sqrt{x}+3}{\sqrt{x^3}-1}=\frac{2\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{2\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{x-10\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{2x+2\sqrt{x}+2+2x-2+x-10\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{5x-8\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{\left(\sqrt{x}-1\right)\left(5\sqrt{x}-3\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{5\sqrt{x}-3}{x+\sqrt{x}+1}\)c) Ta có \(M=\frac{5\sqrt{x}-3}{x+\sqrt{x}+1}\Leftrightarrow Mx+M\sqrt{x}+M-5\sqrt{x}+3=0\Leftrightarrow Mx+\left(M-5\right)\sqrt{x}+\left(M+3\right)=0\)Để phương trình có nghiệm( hay có giá trị x) thì \(\left(M-5\right)^2-4.M.\left(M+3\right)\ge0\Leftrightarrow M^2-10M+25-4M^2-12M\ge0\Leftrightarrow3M^2+22M-25\le0\Leftrightarrow\left(M-1\right)\left(3M+25\right)\le0\Leftrightarrow\)\(-\frac{25}{3}\le M\le1\)

Vậy M có GTLN khi \(\frac{5\sqrt{x}-3}{x+\sqrt{x}+1}=1\Leftrightarrow x+\sqrt{x}+1=5\sqrt{x}-3\Leftrightarrow x-4\sqrt{x}+4=0\Leftrightarrow\left(\sqrt{x}-2\right)^2=0\Leftrightarrow\sqrt{x}-2=0\Leftrightarrow x=4\)

Vậy để biểu thức M có GTLN là 1 thì x=4