To find the absolute maximum and minimum of the function $f(x, y) = 4x + 3y$ on the triangular region bounded by the vertices $(2, 0)$, $(0, 0)$, and $(0, 2)$, we need to evaluate $f(x, y)$ at the vertices and along the edges of the triangle.

Step 1: Evaluate $f(x, y)$ at the vertices

The vertices of the triangle are:

1. $(2, 0)$: $f(2, 0) = 4(2) + 3(0) = 8$
2. $(0, 0)$: $f(0, 0) = 4(0) + 3(0) = 0$
3. $(0, 2)$: $f(0, 2) = 4(0) + 3(2) = 6$

Step 2: Analyze $f(x, y)$ on the edges

Edge 1: $(2, 0)$ to $(0, 0)$ (line segment along $y = 0$)

For $y = 0$, $f(x, y) = 4x$. On this edge, $x$ ranges from $0$ to $2$: $f(x, 0) = 4x, \quad 0 \leq x \leq 2$ The maximum value occurs at $x = 2$ ($f(2, 0) = 8$), and the minimum value occurs at $x = 0$ ($f(0, 0) = 0$).

Edge 2: $(0, 0)$ to $(0, 2)$ (line segment along $x = 0$)

For $x = 0$, $f(x, y) = 3y$. On this edge, $y$ ranges from $0$ to $2$: $f(0, y) = 3y, \quad 0 \leq y \leq 2$ The maximum value occurs at $y = 2$ ($f(0, 2) = 6$), and the minimum value occurs at $y = 0$ ($f(0, 0) = 0$).

Edge 3: $(2, 0)$ to $(0, 2)$ (line segment $y = 2 - x$)

On this edge, $y = 2 - x$, so $f(x, y) = 4x + 3(2 - x)$: $f(x, 2 - x) = 4x + 6 - 3x = x + 6, \quad 0 \leq x \leq 2$ The maximum value occurs at $x = 2$ ($f(2, 0) = 8$), and the minimum value occurs at $x = 0$ ($f(0, 2) = 6$).

Step 3: Determine absolute extrema

• Maximum: The largest value of $f(x, y)$ is $8$, which occurs at $(2, 0)$.
• Minimum: The smallest value of $f(x, y)$ is $0$, which occurs at $(0, 0)$.

Final Answer:

• Absolute Maximum: $f(2, 0) = 8$
• Absolute Minimum: $f(0, 0) = 0$

Would you like a graph or further clarification?

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5 Related Questions:

1. How do you calculate extrema for functions over non-triangular regions?
2. What changes if the function has a quadratic or higher-order term?
3. How does the method adapt for regions bounded by curves instead of lines?
4. Can $f(x, y)$ have an absolute maximum inside the region, and how do we check?
5. How does gradient analysis help in finding extrema?

Tip:

Always check the boundary equations carefully; they determine where extrema may occur!