The vertices of a feasible region are A(1,2), B(5,2), C(1,40. Write a function that satisfies each equation.

a) A is the maximum and B is the minimum.
b) C is the maximum and B is the minimum.
c) B is the maximum and A is the minimum.

To write a function that satisfies each equation, we need to understand the concept of a feasible region and the relationship between the vertices of the region.

A feasible region is the set of all possible solutions to a system of linear inequalities. It is typically represented graphically on the Cartesian plane. The vertices of the feasible region are the points that define the boundaries of the region, and they represent the extreme solutions.

a) A is the maximum and B is the minimum:
In this case, we want to find a linear function that satisfies the condition that point A is the maximum and point B is the minimum. To achieve this, we need to find the line segment that connects points A and B.

We can use the slope-intercept form of a linear equation: y = mx + b.

First, we calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). In this case, (x1, y1) = (1, 2) and (x2, y2) = (5, 2). Substituting these values into the formula, we have m = (2 - 2) / (5 - 1) = 0 / 4 = 0.

Since the line segment is horizontal (slope = 0), the equation becomes y = b, where b is the y-coordinate of either point A or B (since they share the same y-coordinate).

Let's choose point B(5,2) to write the equation:
y = 2.

Therefore, the function that satisfies the condition is f(x) = 2.

b) C is the maximum and B is the minimum:
Similarly, we want to find a linear function that satisfies the condition that point C is the maximum and point B is the minimum. We need to find the line segment that connects points C and B.

Using the slope-intercept form, we calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). In this case, (x1, y1) = (1, 40) and (x2, y2) = (5, 2). Substituting these values into the formula, we have m = (2 - 40) / (5 - 1) = -38 / 4 = -9.5.

The equation becomes y = -9.5x + b, where b is the y-intercept of the line connecting points C and B.

To find b, we plug in the coordinates of either point C or point B. Let's choose point B(5, 2):
2 = -9.5(5) + b
2 = -47.5 + b
b = 49.5.

Therefore, the function that satisfies the condition is f(x) = -9.5x + 49.5.

c) B is the maximum and A is the minimum:
To find a linear function that satisfies the condition that point B is the maximum and point A is the minimum, we determine the line segment that connects points B and A.

Using the slope-intercept form, we calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). In this case, (x1, y1) = (5, 2) and (x2, y2) = (1, 2). Substituting these values into the formula, we have m = (2 - 2) / (1 - 5) = 0 / -4 = 0.

Since the line segment is also horizontal (slope = 0), the equation becomes y = b, where b is the y-coordinate of either point A or B.

Let's choose point A(1, 2) to write the equation:
y = 2.

Therefore, the function that satisfies the condition is f(x) = 2.