Graph the given system and determine the vertices of the overlapping region. Substituting these vertices into the function will help you find the maximum and minimum values.
Graph:
Vertices: (2,-4), (4,-4) Maximum: Does not exist. Minimum: -52
Practice makes perfect
Our first step in finding the maximum and minimum values of the given function is to graph the system and determine the vertices of the overlapping region. Substituting these vertices into the equation, we will find these values.
y≥-3x+2 & (I) 9x+3y≤24 & (II) y≥- 4 & (III)
Inequality I
The equation of the boundary line can be written by changing the inequality symbol in the inequality to an equals sign.
Inequality:& y≥-3x+2
Boundary Line:& y=-3x+2
Since this equation is in slope-intercept form, we can determine its slope m and y-intercept b to draw the line.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= -3x+2
Now that we know the slope and y-intercept, let's use these to draw the boundary line. Notice that the inequality is non-strict, which means that the line will be solid.
To complete the graph, we will test a point that is not on the line and decide which region we should shade. Let's test the point ( 0, 0). If the point satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region.
Since the point does not satisfy the inequality, we will shade the region that does not contain the point.
Inequality II
Let's again start by changing the inequality symbol in the second inequality to an equals sign.
Inequality:& 9x+3y≤24
Boundary Line:& 9x+3y=24
The second step will be to rewrite the equation in the slope-intercept form.
Now, we can determine its slope m and y-intercept b to draw the line.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= -3x+8
Let's use the slope and y-intercept to draw the boundary line. Notice that the inequality is non-strict, which means that the line will be solid.
To complete the graph, we will test a point that is not on the line and decide which region we should shade. Let's test the point ( 0, 0). If the point satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region.
Since the point satisfies the inequality, we will shade the region that contains the point.
Inequality III
The inequality y≥-4 tells us that all coordinate pairs with a y-coordinate that is greater than or equal to -4 will be in the solution set of the inequality. Because the inequality is non-strict, the boundary line will be solid.
Combining the Inequality Graphs
Let's draw the graphs of the inequalities on the same coordinate plane.
Now that we can see the overlapping region, let's highlight the vertices.
Looking at the graph, we can see that the line y=-4 intersects the other lines at the points (2,-4) and (4,-4). These are our vertices. Notice that the feasible region is unbounded. This means that it can go on forever in the positive vertical direction and there are no other vertices.
( 2, -4), and ( 4, -4)
Having all the vertices we can find the maximum and minimum of the given function.
Finding the Maximum and Minimum Values
Since our region is unbounded in one direction, it will have either a maximum or a minimum, but not both. To check which of these it has, we need to calculate the values of the given function at the vertices and at a test point that lies inside the feasible region. Let's use (0,6).
Let's start with calculating the function value in the vertex ( 2, -4).
For the vertex ( 2, -4), the value of the function is -52. We can determine the values of the function for the other points in the same way.
Vertex or Testing Point
f(x,y)=2x+14y
Value
( 2, -4)
f( 2, -4)=2( 2)+14( -4)
f(2,-4)=-52
( 4, -4)
f( 4, -4)=2( 4)+14( -4)
f(4,-4)=-48
( 0, 6)
f( 0, 6)=2( 0)+14( 6)
f(0,6)=84
Looking at the table, we can see that the value of the function in (0,6) is greater than the values in the vertices. That means the maximum value of given function does not exist because the values of this function are increasing as we are moving away from the vertices. However, the minimum value is -52 and it occurs at (2,-4).
