Đáp án:
b. \(Min = - \dfrac{9}{4}\)
Giải thích các bước giải:
\(\begin{array}{l}
a.P = \left[ {\dfrac{{\sqrt a + 1 + \sqrt a - 1}}{{\sqrt a \left( {a - 1} \right)}}} \right]:\dfrac{{2\sqrt a }}{{\sqrt a + 1}}\\
= \dfrac{{2\sqrt a }}{{\sqrt a \left( {a - 1} \right)}}.\dfrac{{\sqrt a + 1}}{{2\sqrt a }}\\
= \dfrac{1}{{\sqrt a \left( {\sqrt a - 1} \right)}} = \dfrac{1}{{a - \sqrt a }}\\
b.Q = \dfrac{1}{P} - 2 = a - \sqrt a - 2\\
= a - 2\sqrt a .\dfrac{1}{2} + \dfrac{1}{4} - \dfrac{9}{4}\\
= {\left( {\sqrt a - \dfrac{1}{2}} \right)^2} - \dfrac{9}{4}\\
Do:{\left( {\sqrt a - \dfrac{1}{2}} \right)^2} \ge 0\forall a \ge 0\\
\to {\left( {\sqrt a - \dfrac{1}{2}} \right)^2} - \dfrac{9}{4} \ge - \dfrac{9}{4}\\
\to Min = - \dfrac{9}{4}\\
\Leftrightarrow \sqrt a - \dfrac{1}{2} = 0\\
\Leftrightarrow a = \dfrac{1}{4}
\end{array}\)
