Consider the following minimum problem: Minimize: \[ C=2 x_{1}+x_{2} \] Subject to the constraints: \[ \left\{\begin{array}{l} 5 x_{1}+x_{2} \geq 9 \\ 2 x_{1}+2 x_{2} \geq 10 \\ x_{1} \geq 0 \\ x_{2} \geq 0 \end{array}\right. \] Write the dual problem for the above minimum problem by selecting the appropriate number for each blank box shown below (Do not solve the dual problem). \[ P=[\text { Select }] \quad \hat{v} y_{1}+[\text { Select }] \quad \hat{v} y_{2} \] [ Select ] \( y_{1}+ \) \( \square \) [ Select ] \( y_{2} \leq 2 \) [ Select ] \( y_{1}+ \) \( \square \) [ Select ] \( y_{2} \leq 1 \) \[ y_{1} \geq 0 \quad ; \quad y_{2} \geq 0 \]

\[ \textbf{Dual Problem:} \] \[ \text{Maximize: } 9y_1 + 10y_2 \] \[ \begin{array}{rcl} 5\,y_{1}+2\,y_{2} & \leq & 2 \quad \text{(from the coefficient of } x_{1}\text{)}\\[1mm] 1\,y_{1}+2\,y_{2} & \leq & 1 \quad \text{(from the coefficient of } x_{2}\text{)} \end{array} \] \[ y_{1}\geq 0,\quad y_{2}\geq 0 \] Thus, filling in the blanks: - In the dual objective function \(P=[\text{Select}]\; \hat{v}y_{1}+[\text{Select}]\; \hat{v}y_{2}\), select “Maximize”, then “9” then “10”. - In the constraint corresponding to \(x_{1}\): fill in “5” for \(y_{1}\) and “2” for \(y_{2}\) so that: \(5y_{1}+2y_{2}\le 2\). - In the constraint corresponding to \(x_{2}\): fill in “1” for \(y_{1}\) and “2” for \(y_{2}\) so that: \(1y_{1}+2y_{2}\le 1\).