Question

optimize the objective function below based on the feasible region shown in the graph above. give both a minimum and a maximum if the region is bound.

objective function: \\(z = 5x + 3y\\)
constraints:
\\(3x + 6y \le 48\\)
\\(5x + 3y \le 45\\)
\\(x \ge 0\\)
\\(y \ge 0\\)

the maximum value is at the point(s)
the minimum value is at the point(s)

Explanation:

Step1: Identify the boundary lines

$$
LATEXBLOCK0
$$

Step2: Find the intersection point

$$
LATEXBLOCK1
$$

Step3: Find all corner points

$$ (0, 0), \quad (9, 0), \quad (0, 8), \quad (6, 5) $$

Step4: Evaluate the objective function

$$
LATEXBLOCK2
$$

Step5: Determine extrema

$$ \text{Maximum: } 45 \text{ at } (9, 0) \text{ and } (6, 5); \quad \text{Minimum: } 0 \text{ at } (0, 0) $$

Answer:

The maximum value is 45 at the point(s) (9, 0)
The minimum value is 0 at the point(s) (0, 0)