Solve the following linear programming problem. Restrict $x \geq 0$ and $y \geq 0$. \[ \begin{array}{c} \text { Maximize } f=3 x+4 y \text { subject to } \\ \begin{array}{r} x+y \leq 11 \\ 2 x+y \leq 16 \\ y \leq 8 \end{array} \\ (x, y)=( \\ f=\square \end{array} \] Submit Answer

Step 1 ：The problem is a linear programming problem. The goal is to maximize the function \(f=3x+4y\) under the constraints \(x+y \leq 11\), \(2x+y \leq 16\), and \(y \leq 8\). We also have the restrictions \(x \geq 0\) and \(y \geq 0\).

Step 2 ：To solve this problem, we need to convert the maximization problem to a minimization problem by multiplying the objective function by -1.

Step 3 ：Then, we can use the linprog function from scipy.optimize to solve the problem.

Step 4 ：The output from the Python code shows that the optimal solution is \(x=3\) and \(y=8\). The optimal value of the objective function is \(f=-(-41)=41\).

Step 5 ：This means that the maximum value of \(f=3x+4y\) under the given constraints is 41, achieved at \((x, y) = (3, 8)\).

Step 6 ：Final Answer: \((x, y) = \boxed{(3, 8)}\), \(f = \boxed{41}\).