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sai đề nha phải là\(\dfrac{a-2ab-b}{2a+3ab-2b}\) nha
ta có \(\dfrac{1}{a}-\dfrac{1}{b}=1\Leftrightarrow\dfrac{b-a}{ab}=1\Leftrightarrow b-a=ab\)
Đặt A=\(\dfrac{a-2ab-b}{2a+3ab-2b}\)
A=\(\dfrac{a-2\left(b-a\right)-b}{2a+3\left(b-a\right)-2b}\) (vì b-a=ab)
A=\(\dfrac{a-2b+2a-b}{2a+3b-3a-2b}\)
A=\(\dfrac{3a-3b}{b-a}=\dfrac{3\left(a-b\right)}{-\left(a-b\right)}=-3\)
Câu 1:
\(Q=a^2+4b^2-10a\)
\(=a^2-10a+25+4b^2-25\)
\(=\left(a-5\right)^2+4b^2-25\)
\(\left(a-5\right)^2\ge0\)
\(4b^2\ge0\)
\(\Rightarrow\left(a-5\right)^2+4b^2-25\ge-25\)
Dấu ''='' xảy ra khi \(\left[\begin{array}{nghiempt}a-5=0\\b=0\end{array}\right.\)
\(\left[\begin{array}{nghiempt}a=5\\b=0\end{array}\right.\)
\(MinQ=-25\Leftrightarrow a=5;b=0\)
Câu 2:
Tam giác DAC vuông tại D có:
\(AC^2=CD^2+AD^2\)
\(=CD^2+CD^2\) (ABCD là hình vuông)
\(=2CD^2\)
\(=2\times\left(3\sqrt{2}\right)^2\)
\(=2\times9\times2\)
\(=36\)
\(AC=\sqrt{36}=6\left(cm\right)\)
Câu 3:
\(\frac{1}{a-1}=1\)
\(a-1=1\)
\(a=1+1\)
\(a=2\)
Thay a = 2 vào P, ta có:
\(P=\frac{2-2\times2\times b-b}{2\times2+3\times2\times b-b}\)
\(=\frac{2-4b-b}{4+6b-b}\)
\(=\frac{2-5b}{4+5b}\)
\(1,Q=\dfrac{a^4-2a^2+a^3-2a+a^2-2}{a^4-2a^2+2a^3-4a+a^2-2}\\ Q=\dfrac{\left(a^2-2\right)\left(a^2+a+1\right)}{\left(a^2-2\right)\left(a^2+2a+1\right)}=\dfrac{a^2+a+1}{a^2+2a+1}\)
\(Q=\dfrac{x^2+x+1}{\left(x+1\right)^2}-\dfrac{3}{4}+\dfrac{3}{4}=\dfrac{x^2+x+1-\dfrac{3}{4}x^2-\dfrac{3}{2}x-\dfrac{3}{4}}{\left(x+1\right)^2}+\dfrac{3}{4}\\ Q=\dfrac{\dfrac{1}{4}x^2-\dfrac{1}{2}x+\dfrac{1}{4}}{\left(x+1\right)^2}+\dfrac{3}{4}=\dfrac{\dfrac{1}{4}\left(x-1\right)^2}{\left(x+1\right)^2}+\dfrac{3}{4}\ge\dfrac{3}{4}\\ Q_{min}=\dfrac{3}{4}\Leftrightarrow x=1\)
\(2,\text{Từ GT }\Leftrightarrow\dfrac{ayz+bxz+czy}{xyz}=0\\ \Leftrightarrow ayz+bxz+czy=0\\ \text{Ta có }\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\\ \Leftrightarrow\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2=1\\ \Leftrightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{xy}{ab}+\dfrac{yz}{bc}+\dfrac{zx}{ca}\right)=0\\ \Leftrightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\cdot\dfrac{cxy+ayz+bzx}{abc}=1\\ \Leftrightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\cdot\dfrac{0}{abc}=1\\ \Leftrightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\)
\(P=\left(a^2+\dfrac{1}{16a^2}\right)+\left(b^2+\dfrac{1}{16b^2}\right)+\dfrac{15}{16}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)\ge2\sqrt{\dfrac{a^2}{16a^2}}+2\sqrt{\dfrac{b^2}{16b^2}}+\dfrac{15}{32}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2\)
\(P\ge1+\dfrac{15}{32}.\left(\dfrac{4}{a+b}\right)^2\ge1+\dfrac{15}{32}.\left(\dfrac{4}{1}\right)^2=\dfrac{17}{2}\)
\(P_{min}=\dfrac{17}{2}\) khi \(a=b=\dfrac{1}{2}\)
\(\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\ge3\sqrt[3]{abc}.3\sqrt[3]{\dfrac{1}{abc}}=9\)
\(\Rightarrow3.P\ge9\Rightarrow P\ge3\)
Dấu "=" xảy ra khi \(a=b=c=1\)
Với \(a,b\in\mathbb{Z};a,b\ne0;a\ne3b;a\ne-5b\), ta có:
\(E=\dfrac{b\left(2a^2+10ab+a+5b\right)}{a-3b}:\dfrac{a^2b+5ab^2}{a^2-3ab}\)
\(=\dfrac{b\left[2a\left(a+5b\right)+\left(a+5b\right)\right]}{a-3b}:\dfrac{ab\left(a+5b\right)}{a\left(a-3b\right)}\)
\(=\dfrac{b\left(2a+1\right)\left(a+5b\right)}{a-3b}:\dfrac{b\left(a+5b\right)}{a-3b}\)
\(=\dfrac{b\left(2a+1\right)\left(a+5b\right)}{a-3b}\cdot\dfrac{a-3b}{b\left(a+5b\right)}\)
\(=2a+1\)
Vì \(2a+1\) là số nguyên lẻ với mọi a nguyên
nên \(E\) là số nguyên lẻ.
\(\text{#}Toru\)
Đề bài thiếu, để tìm min A; B cần thêm điều kiện a;b là số thực dương
Ta có:
\(P=\dfrac{a+3}{a+1}+\dfrac{b+3}{b+1}+\dfrac{c+3}{c+1}\)
\(P=3+2.\left(\dfrac{1}{a+1}+\dfrac{1}{b+1}+\dfrac{1}{c+1}\right)\)
\(P\ge3+2.\dfrac{9}{a+b+c+3}=6\)
Dấu "=" xảy ra \(\Leftrightarrow a=b=c=1\).
Vậy \(min_P=6\), xảy ra khi \(a=b=c=1\)
Áp dụng bđt Cauchy-Schwarz\(A=\dfrac{1}{1+3ab+a^2}+\dfrac{1}{1+3ab+b^2}\)
\(A=\dfrac{1}{1+2ab+ab+a^2}+\dfrac{1}{1+3ab+b^2}\)
\(A\ge\dfrac{\left(1+1\right)^2}{1+2ab+ab+a^2+1+3ab+b^2}\)
\(A\ge\dfrac{4}{\left(a+b\right)^2+4ab+2}=\dfrac{4}{3+4ab}\)
Mặt khác theo AM-GM: \(4ab\le\left(a+b\right)^2\)
\(\Rightarrow\dfrac{4}{3+4ab}\ge\dfrac{4}{3+\left(a+b\right)^2}=\dfrac{4}{3+1}=1\)
\(\Rightarrow A\ge1\)
Dấu "=" xảy ra khi: \(a=b=\dfrac{1}{2}\)
Tính ra là lớp 8 luôn ak