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Lời giải:
$M=\frac{-ab(a-b)}{(a-b)(b-c)(c-a)}+\frac{-bc(b-c)}{(a-b)(b-c)(c-a)}+\frac{-ca(c-a)}{(a-b)(b-c)(c-a)}$
$=\frac{-[ab(a-b)+bc(b-c)+ca(c-a)]}{(a-b)(b-c)(c-a)}$
$=\frac{(ab^2+bc^2+ca^2)-(a^2b+b^2c+c^2a)}{(ab^2+bc^2+ca^2)-(a^2b+b^2c+c^2a)}=1$
a) Ta có : \(a^2+1=a^2+ab+bc+ac=a\left(a+b\right)+c\left(a+b\right)=\left(a+b\right)\left(a+c\right)\)
Tương tự : \(b^2+1=\left(b+a\right)\left(b+c\right)\) ; \(c^2+1=\left(c+a\right)\left(c+b\right)\)
Suy ra \(\left(1+a^2\right)\left(1+b^2\right)\left(1+c^2\right)=\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2\)
Vậy \(A=\frac{\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2}{\left(a+b\right)^2\left(b+c\right)^2\left(c+a\right)^2}=1\)
b) Ta có ; \(a^2+2bc-1=a^2+2bc-\left(ab+bc+ac\right)=a^2-ab+bc-ac=a\left(a-b\right)-c\left(a-b\right)\)
\(=\left(a-b\right)\left(a-c\right)\)
Tương tự : \(b^2+2ac-1=\left(a-b\right)\left(c-b\right)\) ; \(c^2+2ab-1=\left(a-c\right)\left(b-c\right)\)
Suy ra \(\left(a^2+2bc-1\right)\left(b^2+2ac-1\right)\left(c^2+2ab-1\right)=\left(a-b\right)^2.\left(c-a\right)^2.\left[-\left(b-c\right)^2\right]\)
Vậy : \(B=\frac{-\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)}=-1\)
a)Quy đồng hết lên:v
\(=\frac{ab\left(a-b\right)+bc\left(b-c\right)+ca\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{ab\left(a-b\right)-bc\left(a-b+c-a\right)+ca\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{\left(a-b\right)\left(ab-bc\right)+\left(c-a\right)\left(ca-bc\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{b\left(a-b\right)\left(a-c\right)-c\left(a-c\right)\left(a-b\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\) (tắt xíu, ráng hiểu:v)
\(=\frac{\left(a-b\right)\left(a-c\right)\left(b-c\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=-\frac{\left(a-b\right)\left(b-c\right)\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=-1\) (đpcm)
b)(sai thì thôi, cái chỗ đẳng thức xảy ra ý) Đặt \(\frac{a}{b-c}=x;\frac{b}{c-a}=y;\frac{c}{a-b}=z\) (cho nó gọn, viết cho nó lẹ:v) theo câu a) suy ra \(xy+yz+zx=-1\) => \(2xy+2yz+2zx=-2\)
Ta cần chứng minh \(x^2+y^2+z^2\ge2\). Thêm 2xy + 2yz +2zx vào hai vế ta cần chứng minh:
\(x^2+y^2+z^2+2xy+2yz+2zx\ge2+2xy+2yz+2zx\)
\(\Leftrightarrow\left(x+y+z\right)^2\ge2-2=0\) (luôn đúng)
Ta có đpcm. Đẳng thức xảy ra khi \(x+y+z=0\)
Giải
ab + bc + ca = abc =>\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
chọn a = 7 ; b = 3 ; c = \(\frac{21}{11}\)
=> \(\frac{bc}{\left(a+b\right)\left(a+c\right)}+\frac{ca}{\left(b+a\right)\left(b+c\right)}+\frac{ab}{\left(c+a\right)\left(c+b\right)}=0,81>\frac{3}{4}\)
Vậy BĐT phải là :
\(\frac{bc}{\left(a+b\right)\left(a+c\right)}+\frac{ca}{\left(b+a\right)\left(b+c\right)}+\frac{ab}{\left(c+a\right)\left(c+b\right)}\ge\frac{3}{4}\)
quy đồng ta có :
\(\frac{b^2c+bc^2+c^2a+ca^2+a^2b+ab^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\ge\frac{3}{4}\)
<=> 4 .( b2c + bc2 + c2a + ca2 + a2b +ab2 ) \(\ge\)3(2abc + a2b + ab2 + b2c + bc2 + c2a + ca2 )
<=> a2b + ab2 +b2c +bc2 + c2a + ac2 \(\ge\)6abc
<=> \(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge6\)
<=>\(\frac{a+b}{c}+1+\frac{b+c}{a}+\frac{c+a}{b}\ge9\)
<=> \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\) ( 1 )
Ta có BĐT phụ \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{9}{a+b+c}\)
<=> ( a + b + c )( ab + bc + ac ) \(\ge\)9abc
Thật vậy do \(a+b+c\ge3\sqrt[3]{abc}\)
\(ab+bc+ca\ge3\sqrt[3]{a^2b^2c^2}\)
=> \(\left(a+b+c\right)\left(ab+bc+ca\right)\ge9abc\)
\(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge\left(a+b+c\right)\left(\frac{9}{a+b+c}\right)=9\)
đpcm .Dấu " = " xảy ra khi a= b = c
Đề em nghĩ có chút sai sai nên em sửa rồi nha anh ( chắc vậy )
. Ta có: