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\(a^2+b^2=2ab\)
<=> \(a^2+b^2-2ab=0\)
<=> \(\left(a-b\right)^2=0\)
<=> \(a-b=0\)
<=> \(a=b\) (đpcm)
\(a^3+b^3+c^3=3abc\)
<=> \(a^3+b^3+c^3-3abc=0\)
<=> \(\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc=0\)
<=> \(\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)=0\)
<=> \(\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
<=> \(\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{cases}}\)
Xét: \(a^2+b^2+c^2-ab-bc-ca=0\)
<=> \(2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
<=> \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
<=> \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}}\)
<=> \(\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}}\)
<=> \(a=b=c\)
=> đpcm
\(a^4+b^4+c^4\ge\frac{\left(a^2+b^2+c^2\right)\left(a^2+b^2+c^2\right)}{3}\ge\frac{3\sqrt[3]{a^2b^2c^2}\left(a+b+c\right)\left(a+b+c\right)}{9}\)
\(\ge\frac{\sqrt[3]{a^2b^2c^2}.3\sqrt[3]{abc}\left(a+b+c\right)}{3}=abc\left(a+b+c\right)\)
Ta có:\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\forall a,b,c\)
\(\Leftrightarrow a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2\ge0\)
\(\Leftrightarrow2\left(a^2+b^2+c^2\right)-2\left(ab+bc+ca\right)\ge0\)
\(\Leftrightarrow2\left(a^2+b^2+c^2\right)\ge2\left(ab+bc+ca\right)\)
\(\Leftrightarrow a^2+b^2+c^2\ge ab+bc+ca\)
\(\Rightarrow a^2+b^2+c^2\ge2\)
Áp dụng BĐT Cauchy-Schwarz ta có:
\(a^4+b^4+c^4\ge\frac{\left(a^2+b^2+c^2\right)^2}{3}\ge\frac{4}{3}\)
\(\Rightarrow a^4+b^4+c^4\ge\frac{4}{3}\left(đpcm\right)\)
Dấu '=' xảy ra khi\(\hept{\begin{cases}a=b=c\\ab+bc+ca=2\end{cases}\Leftrightarrow a=b=c=\sqrt{\frac{2}{3}}}\)
a: \(4x^2-xy+y^2\)
\(=\left(2x\right)^2-2\cdot2x\cdot\dfrac{1}{4}y+\dfrac{1}{16}y^2+\dfrac{15}{16}y^2\)
\(=\left(2x-\dfrac{1}{4}y\right)^2+\dfrac{15}{16}y^2>=0\)
c: \(a^4+b^4+c^4+d^4\ge4\cdot\sqrt[4]{a^4\cdot b^4\cdot c^4\cdot d^4}=4abcd\)
Áp dụng BĐT Cauchy -Schwarz dạng cộng mẫu thôi:
\(\text{VT}=\frac{1^2}{a}+\frac{1^2}{b}+\frac{2^2}{c}+\frac{4^2}{d}\geq \frac{(1+1+2+4)^2}{a+b+c+d}=\frac{64}{a+b+c+d}=\text{VP}\)
Dấu bằng xảy ra khi \(a=b=\frac{c}{2}=\frac{d}{4}>0\)
1: =>4a^3+4b^3-a^3-3a^2b-3ab^2-b^3>=0
=>a^3-a^2b-ab^2+b^3>=0
=>(a+b)(a^2-ab+b^2)-ab(a+b)>=0
=>(a+b)(a-b)^2>=0(luôn đúng)
2: \(a^4+b^4=\dfrac{a^4}{1}+\dfrac{b^4}{1}>=\dfrac{\left(a^2+b^2\right)^2}{1}=\dfrac{1}{2}\left(\dfrac{a^2}{1}+\dfrac{b^2}{1}\right)^2\)
=>\(a^4+b^4>=\dfrac{1}{2}\left(\dfrac{\left(a+b\right)^2}{2}\right)^2=\dfrac{\left(a+b\right)^4}{8}\)
áp dụng bđt \(\frac{a^2}{x}+\frac{b^2}{y}\ge\frac{\left(a+b\right)^2}{x+y}\)(bđt svacxo) ta có :
VT= \(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\ge\frac{\left(1+1+2+4\right)^2}{a+b+c+d}\)= \(\frac{64}{a+b+c+d}\)=VP (đpcm)
dấu = xảy ra <=>a=b=1; c=2 ; d=4
Dễ dàng CM BĐT phụ sau: \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b},\forall a,b>0\)
Áp dụng liên tục ta có:
\(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\ge\frac{4}{a+b}+\frac{4}{c}+\frac{16}{d}\ge4.\frac{4}{a+b+c}+\frac{16}{d}\ge16.\frac{4}{a+b+c+d}=\frac{64}{a+b+c+d}\)
dấu = xảy ra <=> a+b=c, a+b+c=d, a=b
ĐPCM
Câu a : \(a^2+b^2\ge\dfrac{\left(a+b\right)^2}{2}\Leftrightarrow\left(a-b\right)^2\ge0\)
a: =>2a^2+2b^2>=a^2+2ab+b^2
=>a^2-2ab+b^2>=0
=>(a-b)^2>=0(luôn đúng)
c: =>3a^2+3b^2+3c^2>=a^2+b^2+c^2+2ab+2bc+2ac
=>2a^2+2b^2+2c^2-2ab-2bc-2ac>=0
=>(a-b)^2+(b-c)^2+(a-c)^2>=0(luôn đúng)
Theo BĐT Cauchy ta có :
\(a^4+b^4+c^4+d^4\ge4\sqrt[4]{a^4b^4c^4d^4}=4abcd\)
Dấu ''='' xảy ra khi a = b = c = d = 1