Problem
Minimize: $F=56 x+24 y$ Subject to: \[ \begin{array}{l} x+2 y \geq 14 \\ x+y \geq 12 \\ 2 x+y \geq 16 \\ x, y \geq 0 \end{array} \]
Solution
Step 1 ：We are given the objective function to minimize: \(F=56x+24y\).
Step 2 ：The constraints are: \(x+2y \geq 14\), \(x+y \geq 12\), \(2x+y \geq 16\), and \(x, y \geq 0\).
Step 3 ：We can solve this problem using linear programming.
Step 4 ：The optimal solution is obtained when the value of the objective function is minimum.
Step 5 ：The minimum value of the objective function is 384 and the values of \(x\) and \(y\) that achieve this minimum are \(x=0\) and \(y=16\).
Step 6 ：Final Answer: The minimum value of the function \(F=56x+24y\) subject to the given constraints is 384, achieved at \(x=0\) and \(y=16\). So, the final answer is \(\boxed{384}\).
