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Triển khai vế trái ra, xong chuyển hết sang vế phải ta dc: (a-b)^2+(b-c)^2+(c-a)^2=0
suy ra a-b=0, b-c=0, c-a=0. Vậy a=b=c
Triển khai vế trái ra, xong chuyển hết sang vế phải ta dc: (a-b)^2+(b-c)^2+(c-a)^2=0
suy ra a-b=0, b-c=0, c-a=0. Vậy a=b=c
\(1,\)
Áp dụng BĐT Bunhiacopski:
\(A^2=\left(\sqrt{3-x}+\sqrt{x+7}\right)^2\le\left(1^2+1^2\right)\left(3-x+x+7\right)=2\cdot10=20\)
Dấu \("="\Leftrightarrow3-x=x+7\Leftrightarrow x=-2\)
\(A^2=3-x+x+7+2\sqrt{\left(3-x\right)\left(x+7\right)}\\ A^2=10+2\sqrt{\left(3-x\right)\left(x+7\right)}\ge10\)
Dấu \("="\Leftrightarrow\left(3-x\right)\left(x+7\right)=0\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-7\end{matrix}\right.\)
\(a^3+b^3+c^3-3abc\)
\(=\left(a^3+3a^2b+3ab^2+b^3\right)+c^3-\left(3a^2b+3ab^2+3abc\right)\)
\(=\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(a^2+b^2+2ab-ac-bc+c^2-3ab\right)\)
\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-ac-bc\right)\)\(\left(đpcm\right)\)
\(a^3+b^3+c^3-3abc\)
\(=\left(a+b\right)^3+c^3-3a^2b-3ab^2-3abc\)
\(=\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(\left(a+b\right)^2-c\left(a+b\right)+c^2\right)-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(\left(a+b\right)^2-c\left(a+b\right)+c^2-3ab\right)\)
\(=\left(a+b+c\right)\left(a^2+b^2+2ab-ac-bc+c^2-3ab\right)\)
\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)\)
2)
Xét hiệu:
\(A^2+B^2+C^2+D^2+4-2A-2B-2C-2D\)
\(=\left(A^2-2A+1\right)+\left(B^2-2B+1\right)+\left(C^2-2C+1\right)+\left(D^2-2D+1\right)\)
\(=\left(A-1\right)^2+\left(B-1\right)^2+\left(C-1\right)^2+\left(D-1\right)^2\ge0\)
=> BĐT luôn đúng
Vậy \(A^2+B^2+C^2+D^2+4\ge2\left(A+B+C+D\right)\)
1)
Áp dụng BĐT Cauchy cho 2 số không âm, ta có:
\(\dfrac{AB}{C}+\dfrac{BC}{A}\ge2\sqrt{\dfrac{AB}{C}.\dfrac{BC}{A}}=2B\) (1)
\(\dfrac{BC}{A}+\dfrac{AC}{B}\ge2\sqrt{\dfrac{BC}{A}.\dfrac{AC}{B}}=2C\) (2)
\(\dfrac{AB}{C}+\dfrac{AC}{B}\ge2\sqrt{\dfrac{AB}{C}.\dfrac{AC}{B}}=2A\) (3)
Từ (1)(2)(3) cộng vế theo vế:
\(2\left(\dfrac{AB}{C}+\dfrac{AC}{B}+\dfrac{BC}{A}\right)\ge2\left(A+B+C\right)\)
\(\Rightarrow\dfrac{AB}{C}+\dfrac{AC}{B}+\dfrac{BC}{A}\ge A+B+C\)
Giải:
Ta có:
\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=4\left(a^2+b^2+c^2-ab-bc-ac\right)\)
\(\Leftrightarrow a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2=4a^2+4b^2+4c^2-4ab-4bc-4ac\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=4a^2+4b^2+4c^2-4ab-4bc-4ac\)
\(\Leftrightarrow2\left(a^2+b^2+c^2-ab-bc-ca\right)=4\left(a^2+b^2+c^2-ab-bc-ac\right)\)
\(\Leftrightarrow\left(a^2+b^2+c^2-ab-bc-ca\right)=2\left(a^2+b^2+c^2-ab-bc-ac\right)\)
\(\Rightarrowđpcm\)
biet ab+bc+ca=1CMR (a^2+1)(b^2+1)/(c^2+1)+(b^2+1)(c^2+1)/(a^2+1)+(a^2+1)(b^2+1)/(c^2+1) la binh phuong 1 so huu ti
a) \(\left(a+b+c\right)^2=3\left(ab+bc+ac\right)\)
\(a^2+b^2+c^2+2ab+2ac+2bc-3ab-3ac-3bc=0\)
\(a^2+b^2+c^2-ab-ac-bc=0\)
\(2\left(a^2+b^2+c^2-ab-ac-bc\right)=0\)
\(2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)
\(\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\)
\(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
\(\Rightarrow a=b=c\left(đpcm\right)\)