Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Reasoning An objective function has a maximum value at the vertices \((0,14)\) and \((3,8)\). (a) Can you conclude that it also has a maximum value at the point \((1,12)\) ? Explain. (b) Can you conclude that it also has a maximum value at the point \((4,6)\) ? Explain. (c) Find another point that maximizes the objective function.

Step-by-Step Solution

Answer

The objective function has a maximum value at the point (1,12) as it lies between the vertices (0,14) and (3,8) that are known to be maximum. However, the function does not have a maximum value at the point (4,6) as it does not lie on the line segment between the vertices. Another point that maximizes the objective function could be (2,10), the midpoint between the two vertices (0,14) and (3,8).

Step 1: Determining the Maximum Value at (1,12)

The point (1,12) lies on the line segment between the vertices (0,14) and (3,8). A linear function is a straight line, which means that any point along the line segment between two vertices of maximum values will also yield a maximum value for the linear function. Thus, the point (1,12) also contains a maximum value for the objective function.

Step 2: Determining the Maximum Value at (4,6)

The point (4,6) does not lie on the line segment between the vertices (0,14) and (3,8). Therefore, it does not yield a maximum value for the objective function based on the properties of linear functions. Hence, the objective function does not have a maximum value at the point (4,6).

Step 3: Identifying Another Point That Maximizes the Objective Function

To maximize the objective function, you need to find a point that lies on the line segment between the vertices (0,14) and (3,8). Any point between these two vertices on the line segment will yield a maximum value for the objective function. One such point could be (2,10), which is the midpoint between the two vertices. This point, similar to (1,12), lies on the line segment between (0,14) and (3,8), hence it also be a point of maximum for the objective function.

Key Concepts

Understanding Linear Functions

A linear function is a mathematical representation where the relationship between two variables can be plotted as a straight line on a graph. This can be expressed in the form of the equation: \(y = mx + c\), where \(m\) represents the slope, and \(c\) is the y-intercept.

• The slope \(m\) indicates how steep the line is.
• The y-intercept \(c\) shows where the line crosses the y-axis.

Linear functions are significant because their straightforward nature makes calculations simpler and the relationship between variables is consistent. When dealing with objective functions in optimization problems, this property helps in determining maximum or minimum values with ease.

Finding the Maximum Value

In mathematics, particularly in optimization problems, the maximum value of a function is the highest point along the curve. When dealing with linear functions, especially in terms of an objective function, finding the maximum value directly depends on the function's properties and the points of interest.
In the context of our problem, the function attains its maximum value at the vertices of interest,

• (0,14)
• (3,8)

Any point on the line segment joining these vertices would have a maximum value because linear functions maintain maximum values along the straight path between two known maximum points, assuming there are no constraints altering this path.

The Importance of the Line Segment

A line segment is part of a line that is bounded by two distinct endpoints. When discussing linear functions and optimization, line segments connect points where the function is at its maximum. This was seen in our problem with the points

• (0,14)
• (3,8)

A line segment between these points means any location on this straight path remains optimal. It implies that any point, such as (1,12) or (2,10), along this line segment will still yield a maximum value due to the inherent property of the linear function.
Thus, line segments play a crucial role in visualizing and identifying optimal solutions in graphically represented linear functions.

The Role of Vertices in Finding Extremes

Vertices are critical points often evaluated in identifying extremes like maximum or minimum values in optimization problems. For linear functions, the endpoints of the line segments, known as vertices, play a pivotal role by being potential locations for these extremes.
In scenarios like our exercise, these vertices (

• (0,14)
• (3,8)

) were essential in determining where the maximum values were located. By evaluating the objective function at these vertices and examining the line segment joining them, we conclude that values at these points or anywhere along the path between them are maximum. Thus, understanding vertices' roles helps in efficiently and correctly solving optimization problems.