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Đề viết mệt quá nên thay \(\sqrt{a}=a;\sqrt{b}=b;\sqrt{c}=c\) viết lại đề tiện thể sửa đề luôn.
\(a^2+b^2=\left(a+b-c\right)^2\)
Chứng minh:
\(\frac{a^2+\left(a-c\right)^2}{b^2+\left(b-c\right)^2}=\frac{a-c}{b-c}\)
Ta có: \(a^2+b^2=\left(a+b-c\right)^2\)
\(\Leftrightarrow c^2-2ac-2bc+2ab=0\)
\(\Leftrightarrow a=\frac{c^2-2bc}{2c-2b}\)
Thế vô bài toán ta được
\(VT=\frac{\left(\frac{c^2-2bc}{2c-2b}\right)^2+\left(\frac{c^2-2bc}{2c-2b}-c\right)^2}{b^2+\left(b-c\right)^2}\)
\(=\frac{\left(\frac{c^2-2bc}{2c-2b}\right)^2+\left(\frac{c^2-2bc}{2c-2b}-c\right)^2}{b^2+\left(b-c\right)^2}\)
\(=\frac{\left(\frac{c^2-2bc}{2c-2b}\right)^2+\left(c^2\right)^2}{b^2+\left(b-c\right)^2}=\frac{2c^2\left(2b^2+c^2-2bc\right)}{\left(2b^2+c^2-2bc\right)4\left(c-b\right)^2}=\frac{c^2}{2\left(c-b\right)^2}\)
Ta lại có:
\(VP=\frac{\frac{c^2-2bc}{2c-2b}-c}{b-c}=\frac{-c^2}{-2\left(c-b\right)^2}=\frac{c^2}{2\left(c-b\right)^2}\)
\(\Rightarrow\)ĐOCM
Theo BĐT AM-GM ta có:
\(a+b+c\ge\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)
\(\Rightarrow\left(a+b+c\right)\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)\ge\left(a+b+c\right)^2\)
Mà \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\forall a,b,c\)
\(\Rightarrow\left(a+b+c\right)\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)+\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge\left(a+b+c\right)^2\left(1\right)\)
Do 2 BĐT trên cùng có dấu "=" khi \(a=b=c\)
Dễ dàng theo Cauchy-Schwarz ta có:
\(\left(1^2+1^2+1^2\right)\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\)
\(\Rightarrow3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\left(2\right)\). Giờ cần c/m
\(3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)+\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\)
Mà \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\forall a,b,c\)
Nên cũng chỉ cần chỉ ra
\(3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)\)
Mà \(3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\) (cmt)
\(\Rightarrow\)\(\left(a+b+c\right)^2\)\(\ge\left(a+b+c\right)\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)\)
Dễ thấy \(a+b+c\ne0\) suy ra \(a+b+c\ge\)\(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)
BĐT cuối đúng theo AM-GM (cmt) \((3)\)
Từ \(\left(1\right);\left(2\right);\left(3\right)\) ta có ĐPCM
P/s:bài này liếc phát ra luôn mà quanh đi quẩn lại chỉ mấy BĐT cơ bản :D
C/m lại phần đầu
Cần c/m \((a^2+b^2+c^2)(ab+ac+bc)+\sum_{cyc}(a^2-b^2)^2\geq(a^2+b^2+c^2)^2\)
\(\Leftrightarrow \sum_{cyc}(a^4+a^3b+a^3c-4a^2b^2+a^2bc)\geq0\)
\(\Leftrightarrow \sum_{cyc}(a^4-a^3b-a^3c+a^2bc)+2\sum_{cyc}ab(a-b)^2\geq0\)
Đúng theo Schur
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Đặt \(\sqrt{c.\left(a-c\right)}+\sqrt{c.\left(b-c\right)}\) = A
Ta có A^2 = \(\left(\sqrt{\left(a-c\right).c}+\sqrt{c.\left(b-c\right)}\right)^2\)
Áp dụng bđt bunhiacopxki ta có A^2 <= \(\left(\sqrt{a-c}^2+\sqrt{c^2}\right).\left(\sqrt{c^2}+\sqrt{b-c^2}\right)\)
= (a-c+c).(c+b-c) = ab
<=> A<= \(\sqrt{ab}\)=> ĐPCM
Dấu"=" <=> a-c = c và c = b-c
<=> a=b=2c>0
Ta có bất đẳng thức bunhicopxki
\(\sqrt{ax}+\sqrt{by}\le\sqrt{\left(a+x\right)\left(b+y\right)}\)
Áp dụng vào ta có:
\(\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\le\sqrt{\left(a-c+c\right)\left(b-c+c\right)}\le\sqrt{ab}\)
Dấu bằng xảy ra khi a-c = b-c
lú rùi vậy cũng sai :(
\(BDT\Leftrightarrow\sqrt{\dfrac{c}{b}.\dfrac{a-c}{a}}+\sqrt{\dfrac{c}{a}.\dfrac{b-c}{b}}\le1\)
Áp dụng BĐT AM-GM ta có:
\(VT\le\dfrac{\dfrac{c}{b}+\dfrac{a-c}{a}}{2}+\dfrac{\dfrac{c}{a}+\dfrac{b-c}{b}}{2}=1\)
Bất đẳng thức cần chứng minh tương đương với:
\(\Leftrightarrow\sqrt{\frac{c}{b}\cdot\frac{\left(a-c\right)}{a}}+\sqrt{\frac{c}{a}\cdot\frac{\left(b-c\right)}{b}}\le1\)
Áp dụng Bđt Cô si ta có:
\(\sqrt{\frac{c}{b}\cdot\frac{a-c}{a}}\le\frac{\frac{c}{b}+\frac{a-c}{a}}{2}\left(1\right)\);\(\sqrt{\frac{c}{a}\cdot\frac{b-c}{b}}\le\frac{\frac{c}{a}+\frac{b-c}{b}}{2}\left(2\right)\)
Cộng theo vế của (1) và (2) ta có :Đpcm