Question

cho a = ( 3√x-1)/(√x-2) tìm x thuộc Z để a nhận giá trị nguyên lớn nhất

Answer 1

\(A=\left(3\sqrt{x}-1\right)\left(\sqrt{x}-2\right)\\ A=3x-7\sqrt{x}+2\\ A=\left(\sqrt{3x}\right)^2-2.\sqrt{3x}.\dfrac{7\sqrt{3}}{3}+\dfrac{49}{12}-\dfrac{25}{12}\\ A=\left(\sqrt{3x}-\dfrac{7\sqrt{3}}{3}\right)^2-\dfrac{25}{12}\le-\dfrac{25}{12}\)

Dấu "=" xảy ra khi:

\(\left(\sqrt{3x}-\dfrac{7\sqrt{3}}{3}\right)^2=0\left(x\ge0\right)\\ \Leftrightarrow\sqrt{3x}-\dfrac{7\sqrt{3}}{3}=0\\ \Leftrightarrow\sqrt{3x}=\dfrac{7\sqrt{3}}{3}\\ \Leftrightarrow\sqrt{x}=\dfrac{7}{3}\\ \Leftrightarrow x=\dfrac{49}{9}\left(TM\right)\)

Vậy \(MaxA=-\dfrac{25}{12}\Leftrightarrow x=\dfrac{49}{9}\)