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: (-4,4), (8,4), (6,1), and (2,1) Maximum: 36 Minimum: -36
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.
1≤ y≤ 4 & (I) 4y-6x≥-32 & (II) 2y≥- x+4 & (III)
Inequality I
To graph a compound inequality, we can separate it into two cases.
Compound Inequality:& 1 ≤ y ≤ 4
Case I:& 1 ≤ y
Case II:& y ≤ 4
To make things a bit more simple, we will draw the graph of each case and then combine them.
Case I
The inequality 1 ≤ y describes all values of y that are greater than or equal to 1. This means that every coordinate pair with an y-value that is greater than or equal to 1 will be included in the shaded region. Notice that the inequality is not strict, so the boundary line will be solid.
Case II
Since the inequality represents the values of y that are less than or equal to 4, every coordinate pair with an y-value that is less than or equal to 4 will be included in the shaded region. The boundary line will be solid because the inequality is non-strict.
Combining the Cases
Let's draw the graphs of both cases on the same coordinate plane and shade the overlapping region. This will give us the graph of the compound inequality.
Inequality II
Let's start by changing the inequality symbol in the second inequality to an equals sign.
Inequality:& 4y-6x≥-32
Boundary Line:& 4y-6x=-32
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= 3/2x+(-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 equation of the boundary line can be written by changing the inequality symbol in the inequality to an equals sign.
Inequality:& 2y≥- x+4
Boundary Line:& 2y=- x+4
Let's divide both sides of the equation by 2 to get the equation in the slope-intercept form.
2y=- x+4 ⇒ y=-1/2x+2
Now we can determine its slope m and y-intercept b to draw the line.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= -1/2x+2
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 does not satisfy the inequality, we will shade the region that does not contain the point.
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 write down all four vertices.
( -4, 4), ( 8, 4), ( 6, 1), and ( 2, 1)
Having all the vertices we can find the maximum and minimum of the given function.
Finding the Maximum and Minimum Values
Substituting the vertices into the given function, f(x,y)=-6x+3y, we will determine its maximum and minimum values. Let's start with the vertex ( -4, 4).
For the vertex ( -4, 4), the value of the function is 36. We can determine the values of the function for the other vertices in the same way.
Vertex
f(x,y)=-6x+3y
Value
( -4, 4)
f( -4, 4)=-6( -4)+3( 4)
f(-4,4)=36
( 8, 4)
f( 8, 4)=-6( 8)+3( 4)
f(8,4)=-36
( 6, 1)
f( 6, 1)=-6( 6)+3( 1)
f(6,1)=-33
( 2, 1)
f( 2, 1)=-6( 2)+3( 1)
f(2,1)=-9
Looking at the table, we can see that the maximum value of given function is 36 and it is reached at (-4,4). The minimum value is -36 and it occurs at (8,4).
