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ta có: \(a^2+ab+b^2=\frac{3}{4}\left(a+b\right)^2+\frac{1}{4}\left(a-b\right)^2\)vì (a-b)^2>=0 => \(a^2+ab+b^2\ge\frac{3}{4}\left(a+b\right)^2\Leftrightarrow\sqrt{a^2+ab+b^2}\ge\frac{\sqrt{3}}{2}\left(a+b\right)\)
gọi là A đi. tương tự thì \(A\ge\frac{\sqrt{3}}{2}\left(a+b+b+c+a+c\right)=\frac{\sqrt{3}}{2}.2.1\left(a+b+c=1\right)=\sqrt{3}\Rightarrow MinA=\sqrt{3}\Leftrightarrow a=b=c=\frac{1}{3}\)
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nhận được thông báo thì kéo chuột xuống xem bài giải của t ở phần duyệt bài nhé
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\(a,A=x-4\sqrt{x+9}=\left(x+9-4\sqrt{x+9}+4\right)-13\\ A=\left(\sqrt{x+9}-2\right)^2-13\ge-13\\ A_{min}=-13\Leftrightarrow x+9=4\Leftrightarrow x=-5\\ b,B=x-3\sqrt{x-10}=\left(x-10-3\sqrt{x-10}+\dfrac{9}{4}\right)+\dfrac{31}{4}\\ B=\left(\sqrt{x-10}+\dfrac{9}{4}\right)^2+\dfrac{31}{4}\ge\dfrac{31}{4}\\ B_{min}=\dfrac{31}{4}\Leftrightarrow x-10=\dfrac{81}{16}\Leftrightarrow x=\dfrac{241}{16}\\ c,C=x-\sqrt{x+1}=\left(x+1-\sqrt{x+1}+\dfrac{1}{4}\right)-\dfrac{5}{4}\\ C=\left(\sqrt{x+1}-\dfrac{1}{2}\right)^2-\dfrac{5}{4}\ge-\dfrac{5}{4}\\ C_{min}=-\dfrac{5}{4}\Leftrightarrow x+1=\dfrac{1}{4}\Leftrightarrow x=-\dfrac{3}{4}\)
\(d,D=x+\sqrt{x+2}=\left(x+2+\sqrt{x+2}+\dfrac{1}{4}\right)-\dfrac{9}{4}\\ D=\left(\sqrt{x+2}+\dfrac{1}{4}\right)^2-\dfrac{9}{4}\ge-\dfrac{9}{4}\\ D_{min}=-\dfrac{9}{4}\Leftrightarrow\sqrt{x+2}=-\dfrac{1}{4}\Leftrightarrow x\in\varnothing\)
Vậy dấu \("="\) ko xảy ra
a: \(A=x-4\sqrt{x}+9\)
\(=\left(\sqrt{x}-2\right)^2+5\ge5\forall x\)
Dấu '=' xảy ra khi x=4
b: \(B=x-3\sqrt{x}-10\)
\(=x-2\cdot\sqrt{x}\cdot\dfrac{3}{2}+\dfrac{9}{4}-\dfrac{49}{4}\)
\(=\left(\sqrt{x}-\dfrac{3}{2}\right)^2-\dfrac{49}{4}\ge-\dfrac{49}{4}\forall x\)
Dấu '=' xảy ra khi \(x=\dfrac{9}{4}\)
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By C-S's ine: \(\left(1^2+1^2+1^2\right)\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\)
\(\Rightarrow a^2+b^2+c^2\ge\frac{\left(\frac{3}{2}\right)^2}{3}=\frac{\frac{9}{4}}{3}=\frac{3}{4}\)
Khi \(a=b=c=\frac{1}{2}\)