Với \(a,b,c\) là các số thực dươ...

B

- Hướng dẫn giải

Phương pháp giải:

Giải chi tiết:

Ta có:

\(\begin{array}{l}
\,\,\,\,2\left[ {{{\left( {a + b + c} \right)}^2} - 3\left( {ab + bc + ca} \right)} \right]\\
= 2\left[ {{a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca - 3ab - 3bc - 3ca} \right]\\
= {\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2}\\
\Rightarrow {\left( {a + b + c} \right)^2} - 3\left( {ab + bc + ca} \right) = \frac{1}{2}\left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right]\\
\Rightarrow ab + bc + ca = \frac{{{{\left( {a + b + c} \right)}^2} - \frac{1}{2}\left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right]}}{3}\\
Do\,\,a + b + c = 1 \Rightarrow {\left( {a + b + c} \right)^2} = 1\\
\Rightarrow P = 6\left( {ab + bc + ca} \right) - 2{\left( {a + b + c} \right)^2} + a{\left( {a - b} \right)^2} + b{\left( {b - c} \right)^2} + c{\left( {c - a} \right)^2} + 2\\
\Rightarrow P = 2{\left( {a + b + c} \right)^2} - \left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right] - 2{\left( {a + b + c} \right)^2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + a{\left( {a - b} \right)^2} + b{\left( {b - c} \right)^2} + c{\left( {c - a} \right)^2} + 2\\
\Rightarrow P = - \left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right] + a{\left( {a - b} \right)^2} + b{\left( {b - c} \right)^2} + c{\left( {c - a} \right)^2} + 2\\
\Rightarrow P = {\left( {a - b} \right)^2}\left( {a - 1} \right) + {\left( {b - c} \right)^2}\left( {b - 1} \right) + {\left( {c - a} \right)^2}\left( {c - 1} \right) + 2
\end{array}\)

Do \(a;b;c>0;\,a+b+c=1\Rightarrow 0<a;b;c<1\Rightarrow \left\{ \begin{align}  & a-1

\(\Rightarrow P\le 2\)

Dấu bằng xảy ra \(\Leftrightarrow \left\{ \begin{align}  & a-b=0 \\  & b-c=0 \\  & c-a=0 \\  & a+b+c=1 \\ \end{align} \right.\Leftrightarrow a=b=c=\frac{1}{3}\)

 Chọn B