the minimum and maximum values of the objective function, and the points at which these values occur subject to the given constraints. Round your answers to the rest hundredth. Objective Function Constraints \[ \begin{array}{r} f(x, y)=5 x+8 y \quad x \geq 0 ; y \geq 0 \\ 5 x+y \leq 25 \\ x+3 y \leq 15 \end{array} \]

Step 1 ：First, we need to graph the constraints to find the feasible region. The constraints are \(x \geq 0\), \(y \geq 0\), \(5x + y \leq 25\), and \(x + 3y \leq 15\).

Step 2 ：The feasible region is the area that satisfies all the constraints. In this case, it is the area bounded by the x-axis, the y-axis, the line \(5x + y = 25\), and the line \(x + 3y = 15\).

Step 3 ：Next, we need to find the vertices of the feasible region. These are the points where the boundary lines intersect. The vertices are (0,0), (0,5), (3,4), and (5,0).

Step 4 ：Now, we substitute these vertices into the objective function \(f(x, y) = 5x + 8y\).

Step 5 ：For (0,0), \(f(0, 0) = 5(0) + 8(0) = 0\).

Step 6 ：For (0,5), \(f(0, 5) = 5(0) + 8(5) = 40\).

Step 7 ：For (3,4), \(f(3, 4) = 5(3) + 8(4) = 47\).

Step 8 ：For (5,0), \(f(5, 0) = 5(5) + 8(0) = 25\).

Step 9 ：The minimum value of the objective function is 0 at the point (0,0) and the maximum value is 47 at the point (3,4).

Step 10 ：Finally, we check that these results meet the requirements of the problem. The points (0,0) and (3,4) are in the feasible region, and the values 0 and 47 are the minimum and maximum values of the objective function, respectively.

Step 11 ：So, the minimum value of the objective function is \(\boxed{0}\) at the point \(\boxed{(0,0)}\) and the maximum value is \(\boxed{47}\) at the point \(\boxed{(3,4)}\).