\(\left\{{}\begin{matrix}x\ge3\\E=\sqrt{x-3}+x\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge3\\E=x-3+3+\sqrt{x-3}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge3\\E=\left(\sqrt{x-3}+\dfrac{1}{4}\right)^2+\dfrac{11}{4}\end{matrix}\right.\)
\(\sqrt{x-3}\ge0\Rightarrow\sqrt{x-3}+\dfrac{1}{4}\ge\dfrac{1}{4}\Rightarrow\left(\sqrt{x-3}+\dfrac{1}{4}\right)^2\ge\dfrac{1}{4}\Rightarrow\left(\sqrt{x-3}+\dfrac{1}{4}\right)^2+\dfrac{11}{4}\ge\dfrac{1}{4}+\dfrac{11}{4}=3\)Đẳng thức khi x =3
GTNNE =3 khi x =3
\(x\ge3\)
\(E=x-3+\sqrt{x-3}+\dfrac{1}{4}+\dfrac{11}{4}\)
\(E=\left(\sqrt{x-3}+\dfrac{1}{2}\right)^2+\dfrac{11}{4}\)
\(\sqrt{x-3}\ge0\Rightarrow E\ge\left(\dfrac{1}{2}\right)^2+\dfrac{11}{4}=\dfrac{12}{4}=3\)
GTNN E =3 khi x= 3
