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≤5 & (I) x≤4 & (II) y≥- x & (III)
Inequality I
The inequality y≤5 tells us that all coordinate pairs with a y-coordinate that is less than or equal to 5 will be in the solution set of the inequality. Because the inequality is non-strict, the boundary line will be solid.
Inequality II
Since the inequality represents the values of x that are less than or equal to 4, every coordinate pair with an x-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.
Inequality III
The equation of the boundary line can be written by changing the inequality symbol in the inequality to an equals sign.
Inequality:& y≥- x
Boundary Line:& y=- x
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= -1x+
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 ( 1, 1). 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 see that the line y=5 intersects the lines x=4 and y=- x at the points (4,5) and (-5,5), respectively. These are two of the vertices. The third one, (4,-4), is the point of intersection x=4 and y=- x.
( 4, 5), ( -5, 5), and ( 4, -4)
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-2y, we will determine its maximum and minimum values. Let's start with the vertex ( 4, 5).
For the vertex ( 4, 5), the value of the function is 5. We can determine the values of the function for the other vertices in the same way.
Vertex
f(x,y)=5x-2y
Value
( 4, 5)
f( 4, 5)=5( 4)-2( 5)
f(4,5)=10
( -5, 5)
f( -5, 5)=5( -5)-2( 5)
f(-5,5)=-35
( 4, -4)
f( 4, -4)=5( 4)-2( -4)
f(4,-4)=28
Looking at the table, we can see that the maximum value of given function is 28 and it is reached at (4,-4). The minimum value is -35 and it occurs at (-5,5).
