To find the maximum and minimum values of the objective function [tex]\( F = 4y - 7x \)[/tex] subject to the constraints:
[tex]\[
\begin{array}{l}
y \leq 3x - 1 \\
y \geq -3x + 11 \\
x \leq 6
\end{array}
\][/tex]
we will follow these steps:
### Step 1: Analyze and Visualize Constraints
First, we should identify the feasible region defined by the given constraints:
1. [tex]\( y \leq 3x - 1 \)[/tex]: This constraint defines a region below the line [tex]\( y = 3x - 1 \)[/tex].
2. [tex]\( y \geq -3x + 11 \)[/tex]: This constraint defines a region above the line [tex]\( y = -3x + 11 \)[/tex].
3. [tex]\( x \leq 6 \)[/tex]: This constraint defines a region to the left of the vertical line [tex]\( x = 6 \)[/tex].
### Step 2: Identify the Intersection Points
Next, find the points where these lines intersect to determine the possible corner points of the feasible region:
- Intersection of [tex]\( y = 3x - 1 \)[/tex] and [tex]\( y = -3x + 11 \)[/tex]:
[tex]\[
3x - 1 = -3x + 11 \Rightarrow 6x = 12 \Rightarrow x = 2 \Rightarrow y = 3(2) - 1 = 5 \\
\][/tex]
This gives the point (2, 5).
- Intersection of [tex]\( y = 3x - 1 \)[/tex] and [tex]\( x = 6 \)[/tex]:
[tex]\[
x = 6 \Rightarrow y = 3(6) - 1 = 17 \\
\][/tex]
This gives the point (6, 17).
- Intersection of [tex]\( y = -3x + 11 \)[/tex] and [tex]\( x = 6 \)[/tex]:
[tex]\[
x = 6 \Rightarrow y = -3(6) + 11 = -7 \\
\][/tex]
This gives the point (6, -7).
### Step 3: Determine Feasibility of the Points
We need to confirm which of these points lie within the feasible region bounded by the constraints:
- (2, 5): This point satisfies all three constraints.
- (6, 17): This point violates the second constraint [tex]\( y \geq -3x + 11 \)[/tex].
- (6, -7): This point violates the first constraint [tex]\( y \leq 3x - 1 \)[/tex].
Since feasible points (6, 17) and (6, -7) violate the constraints, the only valid point is (2, 5).
### Step 4: Calculate the Objective Function at Valid Points
Now we calculate the value of the objective function at the feasible point:
[tex]\[
F(2, 5) = 4(5) - 7(2) = 20 - 14 = 6
\][/tex]
### Step 5: Determine Maximum and Minimum Values
Since (2, 5) is the only point in the feasible region, both the maximum and minimum values of the objective function occur at this point:
- Maximum Value: 6
- Minimum Value: 6
Thus, we conclude:
- The minimum value of the objective function [tex]\( F \)[/tex] is 6.
- The maximum value of the objective function [tex]\( F \)[/tex] is 6.
