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.
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≥|x-2| & (I) y≤ 8 & (II) 8y+5x≤49 & (III)
Inequality I
Graphing an absolute value inequality involves two main steps.
Determine which portion of the plane we should shade to show the solution set.
Boundary Line
The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign.
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Inequality&Boundary Line
y ≥ |x-2| & y = |x-2|
The graph of this function is the graph of y=|x| after a horizontal transformation.
Horizontal Translations
Translation right h units, h>0
y=|x- h|
Translation left h units, h>0
y=|x+ h|
Using the table, we can see that there is a horizontal translation 2 units to the right. Applying this transformation to y=|x|, we can draw the boundary line. Because the inequality is not strict, the boundary line will be solid.
Shading the Solution Set
In order to decide which part of the plane to shade, we can test a point which is not on the boundary line. Let's test the point ( 0, 0).
If the point satisfies the inequality, we shade the region that contains the point. Otherwise, we shade the region that does not contain the point.
Since the point does not satisfy the inequality, we will shade the region that does not contain the point.
Inequality II
The inequality y≤8 tells us that all coordinate pairs with a y-coordinate that is less than or equal to 8 will be in the solution set of the inequality. Because the inequality is non-strict, the boundary line will be solid.
Inequality III
Let's again start by changing the inequality symbol in the last inequality to an equals sign.
Inequality:& 8y+5x≤49
Boundary Line:& 8y+5x=49
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= -5/8x+ 49/8
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.
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.
( 2, 0), ( -6, 8), ( -3, 8), and ( 5, 3)
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)=-5x-15y, we will determine its maximum and minimum values. Let's start with the vertex ( 2, 0).
For the vertex ( 2, 0), the value of the function is -10. We can determine the values of the function for the other vertices in the same way.
Vertex
f(x,y)=-5x-15y
Value
( 2, 0)
f( 2, 0)=-5( 2)-15( 0)
f(2,0)=-10
( -6, 8)
f( -6, 8)=-5( -6)-15( 8)
f(-6,8)=-90
( -3, 8)
f( -3, 8)=-5( -3)-15( 8)
f(-3,8)=-105
( 5, 3)
f( 5, 3)=-5( 5)-15( 3)
f(5,3)=-40
Looking at the table, we can see that the maximum value of given function is -10 and it is reached at (2,0). The minimum value is -105 and it occurs at (-3,8).
