Find the minimum and maximum values of the objective function, and the points at which these values occur subject to the given constraints.
Objective Function Constraints
\[
\begin{array}{l}
f(x, y)=4 x+5 y \quad x \geq 0 \\
y \geq 0 \\
x+y \leq 2 \\
\end{array}
\]

Step 1 ：Identify the feasible region. The constraints \(x \geq 0\) and \(y \geq 0\) mean that we are only considering the first quadrant (\(x, y \geq 0\)). The constraint \(x + y \leq 2\) is a line in the xy-plane, and we are considering the region below this line. So, the feasible region is the triangle with vertices at (0,0), (2,0), and (0,2).

Step 2 ：Evaluate the objective function at the vertices. We now evaluate the function \(f(x, y) = 4x + 5y\) at each of the vertices: At (0,0), \(f(0,0) = 4*0 + 5*0 = 0\). At (2,0), \(f(2,0) = 4*2 + 5*0 = 8\). At (0,2), \(f(0,2) = 4*0 + 5*2 = 10\).

Step 3 ：Identify the minimum and maximum values. The minimum value of the function is 0, which occurs at the point (0,0). The maximum value of the function is 10, which occurs at the point (0,2).

Step 4 ：So, the minimum value of the function \(f(x, y) = 4x + 5y\) subject to the given constraints is \(\boxed{0}\) at the point (0,0), and the maximum value is \(\boxed{10}\) at the point (0,2).