To find the maximum and minimum values of the objective function [tex]\( f(x, y) = 10x + 4y \)[/tex] subject to the given constraints, we follow these steps:
1. Identify the Constraints:
[tex]\[
\begin{array}{l}
x \geq 0 \\
y \geq 0 \\
2x + 10y \leq 100 \\
9x + y \leq 54
\end{array}
\][/tex]
2. Determine the Feasible Region:
First, let's write down the equality versions of the constraints to find the boundary lines:
[tex]\[
\begin{aligned}
&2x + 10y = 100 \quad &(i) \\
&9x + y = 54 \quad &(ii)
\end{aligned}
\][/tex]
3. Find the Intersection Points (Vertices):
To find where these lines intersect with each other and the axes:
[tex]\[
\begin{aligned}
& \text{Intersection with} \, x \geq 0 \, \text{and} \, y \geq 0 \, \text{(Axes):} \\
& \text{For} \, x = 0: \, \\
&\quad 2(0) + 10y = 100 \implies y = 10 \\
&\quad 9(0) + y = 54 \implies y = 54 \quad (\text{not in the feasible region}) \\
& \text{For} \, y = 0: \, \\
&\quad 2x + 10(0) = 100 \implies x = 50 \quad (\text{not in the feasible region}) \\
&\quad 9x + 0 = 54 \implies x = 6 \\
\\
& \text{Intersection of the boundary lines:} \\
& \begin{cases}
2x + 10y = 100 \\
9x + y = 54
\end{cases}
\\
& \text{Multiply the second equation by 10 and subtract:} \\
& \quad 20x + 100y = 1000 \\
& \quad 90x + 10y = 540 \\
& \quad 20x + 100y - (90x + 10y) = 1000 - 540 \\
& \quad -70x + 90y = 460 \implies y = 9 - \text{ (substitute back to find x):} \\
\\
& \quad 9(5) + 9 = 54 \implies x = 5, y = 9
\end{aligned}
\][/tex]
4. List the Vertices of the Feasible Region:
- Point (0, 0)
- Point (0, 10)
- Point (5, 9)
- Point (6, 0)
5. Evaluate the Objective Function at Each Vertex:
[tex]\[
\begin{aligned}
&f(0, 0) = 10(0) + 4(0) = 0 \\
&f(0, 10) = 10(0) + 4(10) = 40 \\
&f(5, 9) = 10(5) + 4(9) = 50 + 36 = 86 \\
&f(6, 0) = 10(6) + 4(0) = 60 \\
\end{aligned}
\][/tex]
6. Identify the Maximum and Minimum Values:
- The maximum value is [tex]\( 86 \)[/tex] at the point [tex]\( (5, 9) \)[/tex].
- The minimum value is [tex]\( 0 \)[/tex] at the point [tex]\( (0, 0) \)[/tex].
So, the maximum value of the objective function occurs at [tex]\( (5,9) \)[/tex] with a value of [tex]\( 86 \)[/tex], and the minimum value occurs at [tex]\( (0,0) \)[/tex] with a value of [tex]\( 0 \)[/tex].
