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Ta có \(\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{b}}=2\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{2}{\sqrt{ab}}=4\)
\(\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}=4-\dfrac{2}{\sqrt{ab}}\)
Khi đó P = \(\dfrac{1}{\sqrt{ab}}\left(4-\dfrac{2}{\sqrt{ab}}\right)=-2\left(\dfrac{1}{\sqrt{ab}}-1\right)^2+2\le2\)
Dấu "=" khi a = b = 1
a: \(B=\dfrac{2x+3\sqrt{x}+9-x+3\sqrt{x}}{x-9}=\dfrac{x+9}{x-9}\)
b: \P=A:B
\(=\dfrac{2\sqrt{x}-1}{\sqrt{x}-3}\cdot\dfrac{x-9}{x+9}=\dfrac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}{x+9}>=\dfrac{-1\cdot3}{9}=\dfrac{-1}{3}\)
Dấu = xảy ra khi x=0
\(=\left(1^2+4^2\right)\left(a^2+\dfrac{1}{b^2}\right)\ge\left(1a+4.\dfrac{1}{b}\right)^2\\ \Rightarrow\sqrt{a^2+\dfrac{1}{vb^2}}\ge\dfrac{1}{\sqrt{17}}\left(a+\dfrac{4}{b}\right)\)
Tương tự
\(\sqrt{b^2+\dfrac{1}{c^2}}\ge\dfrac{1}{\sqrt{17}}\left(b+\dfrac{4}{c}\right)\\ \sqrt{c^2+\dfrac{1}{a^2}}\ge\dfrac{1}{\sqrt{17}}\left(c+\dfrac{4}{a}\right)\\ Do.đó:\\ Q\ge\dfrac{1}{\sqrt{17}}\left(a+b+c+\dfrac{4}{a}+\dfrac{4}{b}+\dfrac{4}{c}\right)\ge\dfrac{1}{\sqrt{17}}\\ \left(a+b+c+\dfrac{36}{a+b+c}\right)\)
\(=\dfrac{1}{\sqrt{17}}\\ \left[a+b+c+\dfrac{9}{4\left(a+b+c\right)}+\dfrac{135}{4\left(a+b+c\right)}\right]\\ \ge\dfrac{3\sqrt{17}}{2}\)
Cái thứ nhất là tại sao có cái đầu tiên =)) cái thứ 2 dấu bằng xảy ra khi nào :V
Áp dụng bđt Cauchy - Schwarz ta có:\(Q=\dfrac{2-2a^2b^2}{\left(1+a^2\right)\left(1+b^2\right)}+\dfrac{2}{\sqrt{1+c^2}}=\dfrac{2\left(1-ab\right)\left(1+ab\right)}{\left(ab+bc+ca+a^2\right)\left(ab+bc+ca+b^2\right)}+\dfrac{2}{\sqrt{1+c^2}}=\dfrac{2\left(bc+ca\right)\left(1+ab\right)}{\left(a+b\right)^2\left(b+c\right)\left(c+a\right)}+\dfrac{2}{\sqrt{1+c^2}}=\dfrac{2c\left(1+ab\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\dfrac{2c\left(1+ab\right)}{\sqrt{\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)}}+\dfrac{2}{\sqrt{1+c^2}}\le\dfrac{2c\left(1+ab\right)}{\sqrt{\left(ab+1\right)^2\left(c^2+1\right)}}+\dfrac{2}{\sqrt{1+c^2}}=\dfrac{2c}{\sqrt{c^2+1}}+\dfrac{2}{\sqrt{c^2+1}}=\dfrac{2\left(c+1\right)}{\sqrt{c^2+1}}\le\dfrac{2\left(c+1\right)}{\sqrt{\dfrac{\left(c+1\right)^2}{2}}}=2\sqrt{2}\)Dấu "=" xảy ra khi a = b = \(\sqrt{2}-1;c=1\).
Vậy..
Ta có : \(P=3A+2B\)
\(=\dfrac{2\sqrt{x}}{\sqrt{x}+2}+\dfrac{3}{\sqrt{x}+2}=\dfrac{2\sqrt{x}+3}{\sqrt{x}+2}.\)
\(\Rightarrow P=\dfrac{2\left(\sqrt{x}+2\right)-1}{\sqrt{x}+2}=2-\dfrac{1}{\sqrt{x}+2}\)
Do \(x\ge0\Rightarrow\sqrt{x}+2\ge0\)
\(\Rightarrow-\dfrac{1}{\sqrt{x}+2}\ge-1\)
\(\Rightarrow P=2-\dfrac{1}{\sqrt{x}+2}\ge-1+2=1.\)
Vậy : \(MinP=1.\) Dấu đẳng thức xảy ra khi và chỉ khi \(x=0.\)
\(không\) \(dùng\) \(bđt\) \(làm\) \(sao\) \(ra\) \(được\) ??
\(\sqrt{a^2+\dfrac{1}{b^2}}=\dfrac{1}{\sqrt{17}}.\sqrt{\left(1+4^2\right)\left(a^2+\dfrac{1}{b^2}\right)}\ge\dfrac{1}{\sqrt{17}}\left(a+\dfrac{4}{b}\right)\left(bunhiacopki\right)\)
\(tương-tự:\sqrt{b^2+\dfrac{1}{c^2}}\ge\dfrac{1}{\sqrt{17}}\left(b+\dfrac{4}{c}\right)\)
\(\sqrt{c^2+\dfrac{1}{a^2}}\ge\dfrac{1}{\sqrt{17}}\left(c+\dfrac{4}{a}\right)\)
\(\Rightarrow Q\ge\dfrac{1}{\sqrt{17}}\left(a+b+c+\dfrac{4}{a}+\dfrac{4}{b}+\dfrac{4}{c}\right)=\dfrac{1}{\sqrt{17}}\left[16a+\dfrac{4}{a}+16b+\dfrac{4}{b}+16c+\dfrac{4}{c}-15\left(a+b+c\right)\right]\)
\(bđt:cosi\Rightarrow16a+\dfrac{4}{a}\ge2\sqrt{16a.\dfrac{4}{a}}=2\sqrt{16.4}=16\)
\(tương-tự\Rightarrow16b+\dfrac{4}{b}\ge16;16c+\dfrac{4}{c}\ge16\)
\(có:a+b+c\le\dfrac{3}{2}\Rightarrow15\left(a+b+c\right)\le\dfrac{45}{2}\)
\(\Rightarrow-15\left(a+b+c\right)\ge-\dfrac{45}{2}\)
\(\Rightarrow Q\ge\dfrac{1}{\sqrt{17}}\left(16+16+16-\dfrac{45}{2}\right)=\dfrac{3\sqrt{17}}{2}\)
\(dấu"="xayra\Leftrightarrow a=b=c=\dfrac{1}{2}\)
các bước ban đầu dùng bunhia chọn được 1+4^2 là do dự đoán được trước điểm rơi tại a=b=c=1/2 thôi bạn,cả bước tách dùng cosi cũng dự đoán dc điểm rơi =1/2 nên tách đc thôi
Tại sao lại k được dùng nhỉ? Trông khi dùng thì bài toán sẽ dễ giải quyết hơn
Áp dụng Bunhiacopxki:
\(\sqrt{\left(a^2+\dfrac{1}{b^2}\right)\left(\dfrac{1}{4}+4\right)}\ge\dfrac{a}{2}+\dfrac{2}{b}\)
\(\Rightarrow\sqrt{a^2+\dfrac{1}{b^2}}\ge\dfrac{2}{\sqrt{17}}\left(\dfrac{a}{2}+\dfrac{2}{b}\right)\)
Do đó:
\(Q\ge\dfrac{2}{\sqrt{17}}\left[\dfrac{a+b+c}{2}+2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\right]\)
Ta có: \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{9}{a+b+c}\)
\(\Rightarrow Q\ge\dfrac{2}{\sqrt{17}}\left[\dfrac{a+b+c}{2}+\dfrac{18}{a+b+c}\right]\)
Áp dụng Cô-si:
\(\dfrac{a+b+c}{2}+\dfrac{9}{8\left(a+b+c\right)}\ge\dfrac{3}{2}\)
Do đó:
\(Q\ge\dfrac{2}{\sqrt{17}}\left[\dfrac{3}{2}+\dfrac{135}{8\left(a+b+c\right)}\right]\ge\dfrac{2}{\sqrt{17}}\left[\dfrac{3}{2}+\dfrac{135}{8.\dfrac{3}{2}}\right]=\dfrac{3\sqrt{17}}{2}\)
Dấu "=" xảy ra khi \(a=b=c=\dfrac{1}{2}\)
\(a,P=\dfrac{\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\cdot\dfrac{2-\sqrt{x}}{\sqrt{x}}=\dfrac{-2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+2\right)}=\dfrac{-2}{\sqrt{x}+2}\\ P=-\dfrac{3}{5}\Leftrightarrow\dfrac{2}{\sqrt{x}+2}=\dfrac{3}{5}\\ \Leftrightarrow3\sqrt{x}+6=10\Leftrightarrow\sqrt{x}=\dfrac{4}{3}\Leftrightarrow x=\dfrac{16}{9}\left(tm\right)\)
\(P=-\dfrac{3}{5}\) sao suy ra đc \(\dfrac{2}{\sqrt{x}+2}=\dfrac{3}{5}\) thế
Lời giải:
a. ĐKXĐ: $x>1$
\(B=\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}=\frac{(\sqrt{x+1}+\sqrt{x-1})^2}{2}=x+\sqrt{x^2-1}\)
b.
\(B=\frac{a^2+b^2}{2ab}+\sqrt{\frac{a^2+2ab+b^2}{2ab}.\frac{a^2-2ab+b^2}{2ab}}\)
\(=\frac{a^2+b^2}{2ab}+\sqrt{\frac{(a+b)^2(a-b)^2}{(2ab)^2}}=\frac{a^2+b^2}{2ab}+\frac{|a-b||a+b|}{|2ab|}=\frac{a^2+b^2}{2ab}+\frac{a^2-b^2}{2ab}=\frac{a}{b}\)
c.
$B\leq 1\Leftrightarrow (x-1)+\sqrt{x^2-1}\leq 0$
$\Leftrightarrow \sqrt{x-1}(\sqrt{x-1}+\sqrt{x+1})\leq 0$
$\Leftrightarrow \sqrt{x-1}\leq 0$
Mà $\sqrt{x-1}>0$ với mọi $x<1$ nên điều này vô lý)
Vậy không tồn tại $x$ thỏa đkđb
d.
$B=2\Leftrightarrow x+\sqrt{x^2-1}=2$
$\Leftrightarrow \sqrt{x^2-1}=2-x$
\(\Rightarrow \left\{\begin{matrix} 2-x\geq 0\\ x^2-1=(2-x)^2=x^2-4x+4\end{matrix}\right.\)
\(\Rightarrow x=\frac{5}{4}\)
Thử lại thấy thỏa mãn
Vậy......
\(A=\dfrac{b^2}{b-1}=\dfrac{b^2-1+1}{b-1}=b+1+\dfrac{1}{b-1}=b-1+\dfrac{1}{b-1}+2\)
Áp dụng BĐT cosi cho \(b>0\left(b>1\right)\)
\(A=b-1+\dfrac{1}{b-1}+2\ge2\sqrt{\left(b-1\right)\cdot\dfrac{1}{b-1}}+2=2+2=4\)
Dấu \("="\Leftrightarrow\left(b-1\right)^2=1\Leftrightarrow\left[{}\begin{matrix}b-1=1\\b-1=-1\left(ktm\right)\end{matrix}\right.\Leftrightarrow b=2\left(tm\right)\)