Đáp án:

\[\begin{array}{l}
{B_{\min }} =  - 1\\
{C_{\min }} =  - 36
\end{array}\]

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
*)\\
B = 25{x^2} + 3{y^2} - 10xy + 4y + 1\\
 = \left( {25{x^2} - 10xy + {y^2}} \right) + \left( {2{y^2} + 4y + 2} \right) - 1\\
 = \left[ {{{\left( {5x} \right)}^2} - 2.5x.y + {y^2}} \right] + 2.\left( {{y^2} + 2y + 1} \right) - 1\\
 = {\left( {5x - y} \right)^2} + 2.\left( {{y^2} + 2.y.1 + {1^2}} \right) - 1\\
 = {\left( {5x - y} \right)^2} + 2.{\left( {y + 1} \right)^2} - 1\\
{\left( {5x - y} \right)^2} \ge 0,\,\,\,\forall x,y\\
{\left( {y + 1} \right)^2} \ge 0,\,\,\,\forall y\\
 \Rightarrow B = {\left( {5x - y} \right)^2} + 2.{\left( {y + 1} \right)^2} - 1 \ge  - 1,\,\,\,\forall x,y\\
 \Rightarrow {B_{\min }} =  - 1 \Leftrightarrow \left\{ \begin{array}{l}
{\left( {5x - y} \right)^2} = 0\\
{\left( {y + 1} \right)^2} = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
5x - y = 0\\
y + 1 = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = \dfrac{{ - 1}}{5}\\
y =  - 1
\end{array} \right.\\
 \Rightarrow {B_{\min }} =  - 1\\
*)\\
C = \left( {x + 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - 6} \right)\\
 = \left[ {\left( {x + 1} \right)\left( {x - 6} \right)} \right].\left[ {\left( {x - 2} \right)\left( {x - 3} \right)} \right]\\
 = \left( {{x^2} - 6x + x - 6} \right).\left( {{x^2} - 3x - 2x + 6} \right)\\
 = \left( {{x^2} - 5x - 6} \right)\left( {{x^2} - 5x + 6} \right)\\
 = \left[ {\left( {{x^2} - 5x} \right) - 6} \right].\left[ {\left( {{x^2} - 5x} \right) + 6} \right]\\
 = {\left( {{x^2} - 5x} \right)^2} - {6^2}\\
 = {\left( {{x^2} - 5x} \right)^2} - 36\\
{\left( {{x^2} - 5x} \right)^2} \ge 0,\,\,\,\forall x\\
 \Rightarrow C = {\left( {{x^2} - 5x} \right)^2} - 36 \ge  - 36,\,\,\,\forall x\\
 \Rightarrow {C_{\min }} =  - 36 \Leftrightarrow {\left( {{x^2} - 5x} \right)^2} = 0 \Leftrightarrow {x^2} - 5x = 0\\
 \Leftrightarrow x\left( {x - 5} \right) = 0 \Leftrightarrow \left[ \begin{array}{l}
x = 0\\
x - 5 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = 0\\
x = 5
\end{array} \right.\\
 \Rightarrow {C_{\min }} =  - 36
\end{array}\)