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\(\left(a+b\right)^2\ge4ab\)
\(\Leftrightarrow a^2+2ab+b^2\ge4ab\)
\(\Leftrightarrow a^2+2ab+b^2-4ab\ge0\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\)
\(\Leftrightarrow\left(a-b\right)^2\ge0\left(đpcm\right)\)
Ta có: \(VP=\left(a+b\right)^2-4ab\)
\(=a^2+2ab+b^2-4ab\)
\(=a^2-2ab+b^2\)
\(=\left(a-b\right)^2=VT\)(đpcm)
1) biến đổi vế trái:
= a2+2ab+b2 -a2 +2ab -b2
=4ab = vế phải ( đpcm)
3;5 tuong tu
1) (a + b)2 - (a - b)2 = a2 + 2ab + b2 - a2 + 2ab - b2 = 4ab
3) (a + b)2 - 4ab = a2 + 2ab + b2 - 4ab = a2 - 2ab + b2 = (a - b)2
5) a3 + b3 = a3 + 3a2b + 3ab2 + b3 - 3a2b - 3ab2 = (a + b)3 - 3ab(a + b)
a, \(\left(a+b+c\right)^2=\left[\left(a+b\right)+c\right]^2=\left(a+b\right)^2+2c\left(a+b\right)+c^2=a^2+b^2+c^2+2ab+2ac+2bc\)
b, \(\left(a+b\right)^2+\left(a-b\right)^2=a^2+2ab+b^2+a^2-2ab+b^2=2a^2+2b^2\)
c, \(\left(a+b\right)^2-\left(a-b\right)^2=\left(a+b-a+b\right)\left(a+b+a-b\right)=2b.2a=4ab\)
\(\left(a+b+c\right)^2=\left[\left(a+b\right)+c\right]^2=\left(a+b\right)^2+2\cdot\left(a+b\right)\cdot c+c^2\\ =a^2+2ab+b^2+2ac+2bc+c^2\\ =a^2+b^2+c^2+2ab+2ac+2bc\)
\(\left(a+b\right)^2+\left(a-b\right)^2=a^2+2ab+b^2+a^2-2ab+b^2\\ 2a^2+2b^2\)
\(\left(a+b\right)^2-\left(a-b\right)^2=\left(a+b+a-b\right)\left(a+b-a+b\right)\\ =2a\cdot2b=4ab\)
Ta có :
\(a^2+b^2+1\ge ab+a+b\)
\(\Leftrightarrow a^2+b^2+1-ab-a-b\ge0\)
\(\Leftrightarrow2a^2+2b^2-2ab-2a-2b+2\ge0\)
\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(a^2-2a+1\right)+\left(b^2-2b+1\right)\ge0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(a-1\right)^2+\left(b-1\right)^2\ge0\) ( đúng)
\(VT=\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)
\(=\dfrac{a^2}{ab+ca}+\dfrac{b^2}{ab+bc}+\dfrac{c^2}{ca+bc}\ge\left(Schwarz\right)\dfrac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\)
Mà theo Cô-si ta có:
\(\left\{{}\begin{matrix}a^2+b^2\ge2ab\\b^2+c^2\ge2bc\\c^2+a^2\ge2ca\end{matrix}\right.\Rightarrow a^2+b^2+c^2\ge ab+bc+ca\)
\(\Rightarrow\left(a+b+c\right)^2\ge3\left(ab+bc+ca\right)\) (hằng đẳng thức)
\(\Rightarrow VT\ge\dfrac{3\left(ab+bc+ca\right)}{2\left(ab+bc+ca\right)}=\dfrac{3}{2}\)
Dấu "=" xảy ra khi a=b=c
\(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\)
\(=\frac{a^2}{ab+ac}+\frac{b^2}{ab+bc}+\frac{c^2}{ac+bc}\)
Áp dụng BĐT Cauchy-Schwarz ta có:
\(\frac{a^2}{ab+ac}+\frac{b^2}{ab+bc}+\frac{c^2}{ac+bc}\ge\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\)
Ta c/m BĐT phụ: \(ab+bc+ca\le\frac{1}{3}\left(a+b+c\right)^2\)( b tự c/m nhé. Chuyển vế, c/m VP>=0 là xong )
\(\Rightarrow\frac{a^2}{ab+ac}+\frac{b^2}{ab+bc}+\frac{c^2}{ac+bc}\ge\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\ge\frac{\left(a+b+c\right)^2}{2.\frac{1}{3}\left(a+b+c\right)^2}=\frac{1}{\frac{2}{3}}=\frac{3}{2}\)
\(\Leftrightarrow\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}\)
đpcm
Ta có :
\(\left(a+b\right)^2\ge4ab\)
\(\Leftrightarrow a^2+2ab+b^2\ge4ab\)
\(\Leftrightarrow a^2+2ab+b^2-4ab\ge0\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\)
\(\Leftrightarrow\left(a-b\right)^2\ge0\)( luôn đúng )
\(\left(đpcm\right)\)
Chứng minh phản chứng :
Giả sử : \(\left(a+b\right)^2< 4ab\)
\(\Rightarrow a^2+2ab+b^2< 4ab\)
\(\Rightarrow a^2-2ab+b^2< 0\)
\(\Rightarrow\left(a-b\right)^2< 0\) (vô lí )
Vậy cần có :
\(\left(a+b\right)^2\ge4ab\)