Đáp án:
\[{P_{\min }} = - \frac{1}{2} \Leftrightarrow \left\{ \begin{array}{l}
a = \frac{9}{4}\\
b = \frac{1}{4}
\end{array} \right.\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
P = a - 2\sqrt {ab} + 3b - 2\sqrt a + 1\,\,\,\,\,\,\left( {a,b \ge 0} \right)\\
= 3.\left( {\frac{1}{9}a - \frac{2}{3}\sqrt {ab} + b} \right) + \left( {\frac{2}{3}a - 2\sqrt a + 1} \right)\\
= 3.\left[ {{{\left( {\frac{{\sqrt a }}{3}} \right)}^2} - 2.\frac{{\sqrt a }}{3}.\sqrt b + {{\left( {\sqrt b } \right)}^2}} \right] + \frac{2}{3}.\left( {a - 3\sqrt a + \frac{9}{4}} \right) - \frac{1}{2}\\
= 3.{\left( {\frac{{\sqrt a }}{3} - \sqrt b } \right)^2} + \frac{2}{3}.{\left( {\sqrt a - \frac{3}{2}} \right)^2} - \frac{1}{2} \ge - \frac{1}{2}\\
\Rightarrow {P_{\min }} = - \frac{1}{2} \Leftrightarrow \left\{ \begin{array}{l}
\frac{{\sqrt a }}{3} = \sqrt b \\
\sqrt a - \frac{3}{2} = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = \frac{9}{4}\\
b = \frac{1}{4}
\end{array} \right.
\end{array}\)
