Question

tìm GTNN của \(P=\frac{(1-\sqrt{x})(2+\sqrt{x})}{4-x}\)

Answer 1

Đk: x khác 4

\(P=\dfrac{\left(1-\sqrt{x}\right)\left(2+\sqrt{x}\right)}{4-x}=\dfrac{-x-\sqrt{x}+2}{4-x}=\dfrac{4-x-\sqrt{x}-2}{4-x}=1-\dfrac{\sqrt{x}+2}{4-x}=1-\dfrac{1}{2-\sqrt{x}}\)

Ta có: \(\sqrt{x}\ge0\Leftrightarrow-\sqrt{x}\le0\Leftrightarrow2-\sqrt{x}\le2\Leftrightarrow\dfrac{1}{2-\sqrt{x}}\le\dfrac{1}{2}\Leftrightarrow-\dfrac{1}{2-\sqrt{x}}\ge-\dfrac{1}{2}\Leftrightarrow1-\dfrac{1}{2-\sqrt{x}}\ge1-\dfrac{1}{2}=\dfrac{1}{2}\)

Vậy P đạt GTNN bằng 1/2 <=> x=0