1. **Problem Statement:** We have the linear programming problem with objective function $w = \alpha x_1 + x_2$ and constraints: $$3x_1 + x_2 \leq 9$$ $$x_1 + x_2 \leq 5$$ $$x_1 \geq 0, x_2 \geq 0$$ We want to find the maximum value of $w$ as a function of $\alpha$ by solving graphically. 2. **Step - Identify vertices of feasible region:** The feasible region is bounded by the inequalities and axes. - Intersection with axes: - For $3x_1 + x_2 = 9$: when $x_1=0\Rightarrow x_2=9$, when $x_2=0\Rightarrow x_1=3$ - For $x_1 + x_2 = 5$: when $x_1=0\Rightarrow x_2=5$, when $x_2=0\Rightarrow x_1=5$ - Find intersection of constraints $3x_1 + x_2=9$ and $x_1 + x_2=5$: $$3x_1 + x_2 = 9$$ $$x_1 + x_2 = 5$$ Subtract second from first: $$3x_1 + x_2 - (x_1 + x_2) = 9 - 5 \Rightarrow 2x_1 = 4 \Rightarrow x_1=2$$ Plug back: $$2 + x_2 = 5 \Rightarrow x_2=3$$ 3. **Vertices of feasible polygon:** - $(0,0)$ - $(0,5)$ - $(2,3)$ - $(3,0)$ 4. **Evaluate objective function $w = \alpha x_1 + x_2$ at each vertex:** - At $(0,0)$: $w=0$ - At $(0,5)$: $w=\alpha \cdot 0 + 5 = 5$ - At $(2,3)$: $w=2\alpha + 3$ - At $(3,0)$: $w=3\alpha + 0 = 3\alpha$ 5. **Determine optimal vertex depending on $\alpha$:** Compare values among $(0,5)$, $(2,3)$, and $(3,0)$: - Compare $(0,5)$ and $(2,3)$: $$5 \stackrel{?}{>} 2\alpha + 3 \Rightarrow 2 > 2\alpha \Rightarrow \alpha < 1$$ - Compare $(2,3)$ and $(3,0)$: $$2\alpha + 3 \stackrel{?}{>} 3\alpha \Rightarrow 3 > \alpha \Rightarrow \alpha < 3$$ - Compare $(0,5)$ and $(3,0)$: $$5 \stackrel{?}{>} 3\alpha \Rightarrow \frac{5}{3} > \alpha$$ 6. **Optimal function $w^*(\alpha)$ summarized:** - For $\alpha < 1$, maximum is at $(0,5)$ with $w = 5$ - For $1 \leq \alpha < 3$, maximum is at $(2,3)$ with $w = 2\alpha + 3$ - For $\alpha \geq 3$, maximum is at $(3,0)$ with $w = 3\alpha$ **Answer:** $$w^*(\alpha) = \begin{cases} 5 & \text{if } \alpha < 1 \\ 2\alpha + 3 & \text{if } 1 \leq \alpha < 3 \\ 3\alpha & \text{if } \alpha \geq 3 \end{cases}$$