[Toán 10] Bất đẳng thức

N

nhokdangyeu01

B

braga

[TEX]\blue \fbox{1}. \ \text{Ta co: } \ \ 2(ab+bc+ca)=(a+b+c)^2-(a^2+b^2+b^2) \\ \text{Can CM:} \ a^2+b^2+c^2+2(\sqrt{a}+\sqrt{b}+\sqrt{c})\ge 9 \\ \text{Ma theo AM-GM ta co:} \ \\ a^2+\sqrt{a}+\sqrt{a}\ge 3a \\ b^2+\sqrt{b}+\sqrt{b}\ge 3b \\ c^3+\sqrt{c}+\sqrt{c}\ge 3c \\ \Rightarrow a^2+b^2+c^2+2(\sqrt{a}+\sqrt{b}+\sqrt{c})\ge 3(a+b+c)=9 \\ \text{Tu do suy ra dpcm. Dau bang xay ra}\Leftrightarrow a=b=c=1[/TEX]
 
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