Use the given feasible region determined by the constraint inequalities to find the minimum possible value of the
objective function.
-Find the values of $x$ and $y$ that give the minimum possible value of $f = 6 x + 8 y$ subject to the following constraint $\left\{ \begin{array} { l } 2 x + 4 y \geq 12 \ 2 x + y \geq 8 \ x \geq 0 , y \geq 0 \end{array} \right.$
A) $x = 0 , y = 3$
B) $x = 0 , y = 0$
C) $x = \frac { 10 } { 3 } , y = \frac { 4 } { 3 }$
D) $x = 6 , y = 0$

Question

Quizplus · Accepted Answer

To find the minimum value of the objective function $f = 6x + 8y$ given the constraints, we need to evaluate $f$ at the vertices of the feasible region defined by the inequalities. The feasible region is the area that satisfies all the given inequalities, including $x \geq 0$ and $y \geq 0$, which restricts the region to the first quadrant.1. **Solving the system of inequalities for intersection points (vertices):**- For $2x + 4y = 12$ and $2x + y = 8$, we solve for $x$ and $y$.  - From $2x + 4y = 12$, we can simplify to $x + 2y = 6$.  - From $2x + y = 8$, we leave it as is for now.- Solving these two equations simultaneously gives us the intersection point, which is one of the vertices of the feasible region.- Substituting $y$ from the second equation into the first gives us a system of equations that can be solved to find $x = \frac{10}{3}$ and $y = \frac{4}{3}$.2. **Checking the vertices:**- The vertices of the feasible region are typically where the minimum or maximum of an objective function occurs in linear programming problems.- By evaluating $f = 6x + 8y$ at $(\frac{10}{3}, \frac{4}{3})$, and comparing it with $f$ evaluated at other vertices (which are derived from the intersection points of the inequalities with each other and with the axes), we find that $(\frac{10}{3}, \frac{4}{3})$ gives the minimum value of $f$ within the feasible region.Therefore, the values of $x = \frac{10}{3}$ and $y = \frac{4}{3}$ give the minimum possible value of the objective function $f = 6x + 8y$, making option C correct.

Use the Given Feasible Region Determined by the Constraint Inequalities xxx

Use the Given Feasible Region Determined by the Constraint Inequalities $x$