To evaluate the objective function $C = 20x_1 + 30x_2$ at the given corner points $(0, 0)$, $(2, 0)$, and $(0, 4.71)$, we'll substitute the values of $x_1$ and $x_2$ from each corner point into the equation. This will give us the corresponding values of the objective function at each point, allowing us to determine which one minimizes $C$.

Objective Function:

$C = 20x_1 + 30x_2$

Step-by-Step Evaluation:

1. At Corner Point $(0, 0)$: $C = 20(0) + 30(0) = 0$ So, $C = 0$.

2. At Corner Point $(2, 0)$: $C = 20(2) + 30(0) = 40 + 0 = 40$ So, $C = 40$.

3. At Corner Point $(0, 4.71)$: $C = 20(0) + 30(4.71) = 0 + 141.3 = 141.3$ So, $C = 141.3$.

Conclusion:

The value of $C$ is the smallest at the corner point $(0, 0)$, where $C = 0$. Therefore, the minimum value of the objective function is $C = 0$, and the corresponding corner point is $(0, 0)$.

Would you like further details or explanations on any part of this process?

Here are 5 related questions for further study:

1. How do you interpret the values of the objective function in terms of optimization?
2. What if there were more corner points; how would you handle those in the evaluation?
3. What are the constraints that could influence the choice of corner points in a linear programming problem?
4. How do you determine if a given solution is feasible in linear programming problems?
5. How would you solve a linear programming problem using graphical methods?

Tip: Always ensure to check if the corner points fall within the feasible region of the constraints before evaluating the objective function.