a) \(b^2=ac.CMR:\frac{a^2+b^2}{b^2+c^2}=\frac{a}{c}\)
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\(b^2=ac.CMR:\frac{a^2+b^2}{b^2+c^2}=\frac{a}{c}\)
Ta có:
b^2=ac \(\Rightarrow\) \(\frac{a}{b}\)=\(\frac{b}{c}\)\(\Rightarrow\) \(\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{ab}{bc}=\frac{a}{c}\)(1)
Áp dụng tính chất của dãy tỉ số = nhau ta có:
\(\frac{a}{b}=\frac{a}{c}=\frac{a+b}{b+c}\)
\(\Rightarrow\)\(\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{a^2+b^2}{b^2+c^2}\)(2)
Từ (1) và (2)\(\Rightarrow\)đpcm
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Cho :\(b^2=ac.CMR:\frac{a^2+b^2}{b^2+c^2}=\frac{a}{c}\)
xin bà con cô bác tick cho mik nghen
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olm nói ko được trả lời câu hỏi bằng chtt hoặc bảo người khác tick cho mình là bị trừ điểm hỏi đáp đó các bn
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\(Chob^2=ac.CMR:\frac{a^2+b^2}{b^2+c^2}=\frac{a}{c}\)
\(b^2=ac\Leftrightarrow\frac{a}{b}=\frac{b}{c}=k\Leftrightarrow\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{a^2+b^2}{b^2+c^2}=k^2\)
mà a =bk ; b = ck => a =c k2 => k2 =a/c
=>\(\frac{a^2+b^2}{b^2+c^2}=k^2=\frac{a}{c}\)
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Cho a, b, c > 0. CM:
a) \(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\)
b) \(\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{a^2+c^2}{a+c}\le\frac{3\left(a^2+b^2+c^2\right)}{a+b+c}\)
c) \(\frac{a^2+b^2}{a^2-2ab+b^2}+\frac{b^2+c^2}{b^2-2bc+c^2}+\frac{c^2+a^2}{c^2-2ac+a^2}\ge\frac{5}{2}\)
(a, b, c đôi một khác nhau)
áp dụng cô si ta có:+)frac{a^5}{b^3}+frac{a^3}{b}gefrac{2a^4}{b^2};frac{b^5}{c^3}+frac{b^3}{c}gefrac{2b^4}{c^2};frac{c^5}{a^3}+frac{c^3}{a}gefrac{2c^4}{a^2}Leftrightarrowfrac{a^5}{b^3}+frac{b^5}{c^3}+frac{c^5}{a^3}ge2left(frac{a^4}{b^2}+frac{b^4}{c^2}+frac{c^4}{a^2}right)-left(frac{a^3}{b}+frac{b^3}{c}+frac{c^3}{a}right)+)frac{a^4}{b^2}+a^2gefrac{2a^3}{b};frac{b^4}{c^2}+b^2gefrac{2b^3}{c};frac{c^4}{a^2}+c^2gefrac{2C^3}{a}Leftrightarrowfrac{a^4}{b^2}+frac{b^4}{c^2}+frac{c^4}{a^2}ge2left(frac{a^3}...
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áp dụng cô si ta có:
+)\(\frac{a^5}{b^3}+\frac{a^3}{b}\ge\frac{2a^4}{b^2};\frac{b^5}{c^3}+\frac{b^3}{c}\ge\frac{2b^4}{c^2};\frac{c^5}{a^3}+\frac{c^3}{a}\ge\frac{2c^4}{a^2}\)
\(\Leftrightarrow\frac{a^5}{b^3}+\frac{b^5}{c^3}+\frac{c^5}{a^3}\ge2\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)-\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)\)
+)\(\frac{a^4}{b^2}+a^2\ge\frac{2a^3}{b};\frac{b^4}{c^2}+b^2\ge\frac{2b^3}{c};\frac{c^4}{a^2}+c^2\ge\frac{2C^3}{a}\)
\(\Leftrightarrow\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\ge2\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)-\left(a^2+b^2+c^2\right)\)
+)\(\frac{a^3}{b}+ab\ge2a^2;\frac{b^3}{c}+bc\ge2b^2;\frac{c^3}{a}+ca\ge2c^2\)
\(\Leftrightarrow\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\ge\left(a^2+b^2+c^2\right)+\left(a^2+b^2+c^2-ab-bc-ca\right)\ge\left(a^2+b^2+c^2\right)\)
\(\Leftrightarrow\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\ge\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)+\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-a^2-b^2-c^2\right)\ge\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\)
\(\Leftrightarrow\frac{a^5}{b^3}+\frac{b^5}{c^3}+\frac{c^5}{a^3}\ge\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)+\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}-\frac{a^3}{b}-\frac{b^3}{c}-\frac{c^3}{a}\right)\ge\left(\frac{a^4}{b^2}+\frac{b^4}{c^2}+\frac{c^4}{a^2}\right)\)
vì a+b+c0 nên a-(b+c)Rightarrow $a^2$$(b+c)^2$tương tự ta có : $b^2$$(a+c)^2$$c^2$$(a+b)^2$Rightarrow $frac{a^2}{a^2-b^2-c^2}$+$frac{b^2}{b^2-c^2-a^2}$+$frac{c^2}{c^2-b^2-a^2}$$frac{a^2}{(b+c)^2-b^2-c^2}$+$frac{b^2}{(a+c)^2-a^2-c^2}$+$frac{c^2}{(a+b)^2-a^2-b^2}$$frac{a^2}{2bc}$+$frac{b^2}{2ac}$+$frac{c^2}{2ab}$$frac{a^3+b^3+c^3}{2abc}$vì a+b+c0 nên a^3+b^3+c^33abc(hằng đẳng thức nâng cao)Rightarrow $frac{a^3+b^3+c^3}{2abc}$$frac{3}{2}$
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vì a+b+c=0 nên a=-(b+c)\Rightarrow $a^2$=$(b+c)^2$
tương tự ta có : $b^2$=$(a+c)^2$
$c^2$=$(a+b)^2$
\Rightarrow $\frac{a^2}{a^2-b^2-c^2}$+$\frac{b^2}{b^2-c^2-a^2}$+$\frac{c^2}{c^2-b^2-a^2}$
=$\frac{a^2}{(b+c)^2-b^2-c^2}$+$\frac{b^2}{(a+c)^2-a^2-c^2}$
+$\frac{c^2}{(a+b)^2-a^2-b^2}$
=$\frac{a^2}{2bc}$+$\frac{b^2}{2ac}$+$\frac{c^2}{2ab}$
=$\frac{a^3+b^3+c^3}{2abc}$
vì a+b+c=0 nên a^3+b^3+c^3=3abc(hằng đẳng thức nâng cao)
\Rightarrow $\frac{a^3+b^3+c^3}{2abc}$=$\frac{3}{2}$
a) \(A=\left(1+\frac{b^2+c^2-a^2}{2bc}\right).\frac{1+\frac{a}{b+c}}{1-\frac{a}{b+c}}.\frac{b^2+c^2-\left(b-c\right)^2}{a+b+c}\)
b) \(B=\frac{\frac{3a}{a+b}}{\frac{2a}{a^2-2ab+b^2}}\)
c) \(C=\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}:\frac{\frac{a^2}{b^2}-\frac{b^2}{a^2}}{\left(\frac{1}{a}+\frac{1}{b}\right)^2}\)
a) \(A=\left(1+\frac{b^2+c^2-a^2}{2bc}\right).\frac{1+\frac{a}{b+c}}{1-\frac{a}{b+c}}.\frac{b^2+c^2-\left(b-c\right)^2}{a+b+c}\)
\(=\frac{2bc+b^2+c^2-a^2}{2bc}.\frac{\frac{a+b+c}{b+c}}{\frac{b+c-a}{b+c}}.\frac{b^2+c^2-b^2+2bc-c^2}{a+b+c}\)
\(=\frac{\left(b+c+a\right)\left(b+c-a\right)}{2bc}.\frac{a+b+c}{b+c-a}.\frac{2bc}{a+b+c}\)
\(=a+b+c\)
b) \(B=\frac{\frac{3a}{a+b}}{\frac{2a}{a^2-2ab+b^2}}\)\(=\frac{3a}{a+b}.\frac{\left(a-b\right)^2}{2a}=\frac{3\left(a-b\right)^2}{2\left(a+b\right)}\)
c) \(C=\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}:\frac{\frac{a^2}{b^2}-\frac{b^2}{a^2}}{\left(\frac{1}{a}+\frac{1}{b}\right)^2}\)
\(=\frac{\frac{a^2+b^2}{ab}}{\frac{a^2-b^2}{ab}}:\frac{\frac{a^4-b^4}{a^2b^2}}{\frac{\left(a+b\right)^2}{a^2b^2}}\)
\(=\frac{a^2+b^2}{a^2-b^2}.\frac{\left(a+b\right)^2}{a^4-b^4}\)
\(=\frac{\left(a^2+b^2\right)\left(a+b\right)^2}{\left(a+b\right)\left(a-b\right)\left(a^2+b^2\right)\left(a+b\right)\left(a-b\right)}\)
\(=\frac{1}{\left(a-b\right)^2}\)
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Leftrightarrowfrac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b}+frac{ab}{a+b}+frac{bc}{b+c}+frac{ca}{c+a}ge a+b+cLeftrightarrowfrac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b}+a-frac{a^2}{a+b}+b-frac{b^2}{b+c}+c-frac{c^2}{c+a}ge a+b+cLeftrightarrowfrac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b}gefrac{a^2}{a+b}+frac{b^2}{b+c}+frac{c^2}{c+a}Leftrightarrow a^2left(a+bright)left(a+cright)+b^2left(b+aright)left(b+cright)+c^2left(c+aright)left(c+bright)ge a^2left(a+cright)left(b+cright)+b^2left(b+aright)left(c+ar...
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\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\ge a+b+c\)
\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+a-\frac{a^2}{a+b}+b-\frac{b^2}{b+c}+c-\frac{c^2}{c+a}\ge a+b+c\)
\(\Leftrightarrow\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\)
\(\Leftrightarrow a^2\left(a+b\right)\left(a+c\right)+b^2\left(b+a\right)\left(b+c\right)+c^2\left(c+a\right)\left(c+b\right)\ge a^2\left(a+c\right)\left(b+c\right)+b^2\left(b+a\right)\left(c+a\right)+c^2\left(c+b\right)\left(a+b\right)\)
\(\Leftrightarrow a^4+b^4+c^4\ge a^2c^2+a^2b^2+b^2c^2\left(lđ\right)\)
\(\Leftrightarrow\frac{a^2+bc}{b+c}+\frac{b^2+ca}{c+a}+\frac{c^2+ab}{a+b}\ge a+b+c\)
cho b^2=ac.cmr a^2+b^2/b^2+c^2=2/c
\(=\frac{2}{c}\)clgt ?
Phải là \(\frac{a}{c}\)chứ
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Ta có: b2 = ac
<=> a/b = b/c
<=> a2/b2 = b2/c2 = (a2 + b2)/(b2 + c2) (1)
Lại có: a2/b2 = a/b . a/b = a/b . b/c = a/c (2)
Từ (1) (2) => (a2 + b2)/(b2 + c2) = a/c
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