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A.
$a^2+4b^2+9c^2=2ab+6bc+3ac$
$\Leftrightarrow a^2+4b^2+9c^2-2ab-6bc-3ac=0$
$\Leftrightarrow 2a^2+8b^2+18c^2-4ab-12bc-6ac=0$
$\Leftrightarrow (a^2+4b^2-4ab)+(a^2+9c^2-6ac)+(4b^2+9c^2-12bc)=0$
$\Leftrightarrow (a-2b)^2+(a-3c)^2+(2b-3c)^2=0$
$\Rightarrow a-2b=a-3c=2b-3c=0$
$\Rightarrow A=(0+1)^{2022}+(0-1)^{2023}+(0+1)^{2024}=1+(-1)+1=1$
B.
$x^2+2xy+6x+6y+2y^2+8=0$
$\Leftrightarrow (x^2+2xy+y^2)+y^2+6x+6y+8=0$
$\Leftrightarrow (x+y)^2+6(x+y)+9+y^2-1=0$
$\Leftrightarrow (x+y+3)^2=1-y^2\leq 1$ (do $y^2\geq 0$ với mọi $y$)
$\Rightarrow -1\leq x+y+3\leq 1$
$\Rightarrow -4\leq x+y\leq -2$
$\Rightarrow 2020\leq x+y+2024\leq 2022$
$\Rightarrow A_{\min}=2020; A_{\max}=2022$
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\(Q=a^2+4b^2+10a=a^2+2.a.5+25-25+\left(2b\right)^2=\left(a+5\right)^2+\left(2b\right)^2+\left(-25\right)\)
\(\left(a+5\right)^2\ge0;\left(2b\right)^2\ge0=>\left(a+5\right)^2+\left(2b\right)^2\ge0=>\left(a+5\right)^2+\left(2b\right)^2+\left(-25\right)\ge0\)
Vậy GTNN của Q là - 25. Dấu "=" xảy ra khi a + 5 = 0 => a = -5 và 2b = 0 => b = 0
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\(Q=a^2-10a+25-25+4b^2\)
\(Q=\left(a^2-2.5.a+5^2\right)+4b^2-25=\left(a-5\right)^2+4b^2-25\)
\(Q\ge-25\) đẳng thức khi \(\hept{\begin{cases}a=5\\b=0\end{cases}}\)
Q=a2+4b2-10a
=a2-10a+25-25+4b2
=(a-5)2+4b2-25
\(\Rightarrow\left(a-5\right)^2+4b^2\ge0\) voi moi a
\(\Leftrightarrow\left(a-5\right)^2+4b^2\ge-25\)
Vay GTNN la -25
Dau "=" xay ra khi : a-5=0 \(\Rightarrow\)a=5
4b=0 \(\Rightarrow\)b=0
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C=2a2+b2-2ab+10a+42
=a2-2ab+b2+a2+10a+25+17
=(a-b)2+(a+5)2+17
=>MIN(C)=17 <=>a-b=0 và a+5=0
<=>a=b=-5
vậy ..................
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\(A=\dfrac{5x^2}{x^2}-\dfrac{x}{x^2}+\dfrac{1}{x^2}=\dfrac{1}{x^2}-\dfrac{1}{x}+5=\left(\dfrac{1}{x^2}-\dfrac{1}{x}+\dfrac{1}{4}\right)+\dfrac{19}{4}=\left(\dfrac{1}{x}-\dfrac{1}{2}\right)^2+\dfrac{19}{4}\ge\dfrac{19}{4}\)
\(A_{min}=\dfrac{19}{4}\) khi \(\dfrac{1}{x}=\dfrac{1}{2}\Rightarrow x=2\)
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Bài 1:
\(A=x^2+6x+9+x^2-10x+25\)
\(=2x^2+4x+34\)
\(=2\left(x^2+2x+17\right)\)
\(=2\left(x+1\right)^2+32>=32\forall x\)
Dấu '=' xảy ra khi x=-1
a2 + 4b2 - 10a = (a2 - 10a + 25) + 4b2 - 25
= (a - 5)2 + 4b2 - 25\(\ge25\)
Sai rồi cái này nhỏ nhất phải là -25 chứ