The, objective function is maximum or minimum at a point, which lies on the boundary of the feasible region.

Answer

To solve the problem step by step, we will analyze the given objective function and constraints, graph the feasible region, identify the corner points, and evaluate the objective function at these points to find the maximum and minimum values.

### Step-by-Step Solution:

1. **Define the Objective Function and Constraints:**
- The objective function is given as \( z = 2x + 4y \).
- The constraints are:
- \( x + y \leq 20 \)
- \( x \geq 0 \)
- \( y \geq 0 \)

2. **Graph the Constraints:**
- Start by graphing the line \( x + y = 20 \). This line intersects the axes at points (20, 0) and (0, 20).
- Since \( x + y \leq 20 \), shade the area below this line, which represents the feasible region.
- The constraints \( x \geq 0 \) and \( y \geq 0 \) indicate that we are only interested in the first quadrant (where both x and y are non-negative).

3. **Identify the Feasible Region:**
- The feasible region is bounded by the axes and the line \( x + y = 20 \). The corner points of this region are:
- (0, 0)
- (20, 0)
- (0, 20)

4. **Evaluate the Objective Function at Corner Points:**
- Calculate \( z \) at each corner point:
- At (0, 0):
\[ z = 2(0) + 4(0) = 0 \]
- At (20, 0):
\[ z = 2(20) + 4(0) = 40 \]
- At (0, 20):
\[ z = 2(0) + 4(20) = 80 \]

5. **Determine Maximum and Minimum Values:**
- From the calculations:
- The maximum value of \( z \) is 80 at the point (0, 20).
- The minimum value of \( z \) is 0 at the point (0, 0).

6. **Conclusion:**
- The objective function \( z \) reaches its maximum and minimum values at the corner points of the feasible region, confirming that the objective function is maximum or minimum at points lying on the boundary of the feasible region.