Question

Tìm gtnn của biểu thức: K=\(\sqrt{5x+6\sqrt{5x-9}}+\sqrt{5x-6\sqrt{5x-9}}\)

Answer 1

\(K=\sqrt{5x-9+6\sqrt{5x-9}+9}+\sqrt{5x-9-6\sqrt{5x-9}+9}\)

\(=\sqrt{\left(\sqrt{5x-9}+3\right)^2}+\sqrt{\left(3-\sqrt{5x-9}\right)^2}\)

\(=\left|\sqrt{5x-9}+3\right|+\left|3-\sqrt{5x-9}\right|\ge\left|\sqrt{5x-9}+3+3-\sqrt{5x-9}\right|=6\)

\(K_{min}=6\) khi \(\left\{{}\begin{matrix}5x-9\ge0\\3-\sqrt{5x-9}\ge0\end{matrix}\right.\) \(\Rightarrow\dfrac{9}{5}\le x\le\dfrac{18}{5}\)