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a: \(P=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2=x-\sqrt{x}+1\)
b: \(P=\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\forall x\)
Dấu '=' xảy ra khi x=1/4
\(A=\left(x-2\sqrt{xy}+y\right)\)\(-\left(2\sqrt{x}-2\sqrt{y}\right)\)\(+1\)\(+\left(2y-2\sqrt{y}+\frac{1}{2}\right)\)\(-\frac{1}{2}\)
\(=\left(\sqrt{x}-\sqrt{y}\right)^2-2\left(\sqrt{x}-\sqrt{y}\right)\)\(+1\)\(+2\left(y-\sqrt{y}+\frac{1}{4}\right)+\frac{1}{2}\)
\(\left(\sqrt{x}-\sqrt{y}-1\right)^2\)\(+2\left(\sqrt{y}-\frac{1}{2}\right)^2+\frac{1}{2}\)lớn hơn hoặc bằng \(\frac{1}{2}\)
A min \(=\frac{1}{2}\)<=>\(\left(\sqrt{x}-\sqrt{y}-1\right)^2\)=0, \(\left(\sqrt{y}-\frac{1}{2}\right)^2=0\)<=> \(x=\frac{9}{4};y=\frac{1}{4}\).
\(M=2x^2-8x+\sqrt{x^2-4x+5}+6\)
\(=2\left(x^2-4x+5\right)+\sqrt{x^2-4x+5}-4\)
Đặt \(\sqrt{x^2-4x+5}=t\)
Ta thấy \(x^2-4x+5=\left(x^2-4x+4\right)+1=\left(x+2\right)^2+1\ge1\)
Vậy nên \(\sqrt{x^2-4x+5}\ge1\Rightarrow t\ge1\)
Khi đó \(M=2t^2+t-4=2\left(t^2+\frac{1}{2}t-2\right)=2\left[\left(t^2+2.t.\frac{1}{4}+\frac{1}{16}\right)-\frac{33}{16}\right]\)
\(=2\left[\left(t+\frac{1}{4}\right)^2-\frac{33}{16}\right]=2\left(t+\frac{1}{4}\right)^2-\frac{33}{8}\)
Do \(t\ge1,\left(t+\frac{1}{4}\right)^2\ge\frac{25}{16}\)
Vậy thì \(M\ge2.\frac{25}{16}-\frac{33}{8}=-1\)
Vậy \(minM=-1\) khi t = 1
hay \(\sqrt{x^2-4x+5}=0\Rightarrow x^2-4x+5=2\Rightarrow x^2-4x+4=0\Rightarrow x=2\)
\(x^2+x\sqrt{3}+1\)
\(=x^2+2.x.\frac{\sqrt{3}}{2}+\left(\frac{\sqrt{3}}{2}\right)^2+\frac{1}{4}\)
\(=\left(x+\frac{\sqrt{3}}{2}\right)^2+\frac{1}{4}\ge\frac{1}{4}\)
Dấu "=" xảy ra khi \(x=-\frac{\sqrt{3}}{2}\)
Đặt \(A=x^2+x\sqrt{3}+1\)
\(\Rightarrow A=x^2+x\sqrt{3}+\frac{3}{4}+\frac{1}{4}\)
\(\Rightarrow A=\left(x+\frac{\sqrt{3}}{2}\right)^2+\frac{1}{4}\)
Vì \(\left(x+\frac{\sqrt{3}}{2}\right)^2\ge0\forall x\Rightarrow\)\(\left(x+\frac{\sqrt{3}}{2}\right)^2+\frac{1}{4}\ge\frac{1}{4}\forall x\)
Vậy \(A_{min}=\frac{1}{4}\Leftrightarrow x+\frac{\sqrt{3}}{2}=0\Leftrightarrow x=-\frac{\sqrt{3}}{2}\)