We are asked to find the absolute minimum and maximum values of the function $f(x, y) = 5x + 2y$ on the closed region where $-3 \leq x \leq 3$ and $-3 \leq y \leq 3$.

Step-by-Step Approach:

1. The function $f(x, y) = 5x + 2y$ is linear, meaning that it doesn't have any critical points inside the region (since the partial derivatives don't depend on $x$ or $y$). This implies the extreme values will occur on the boundary of the region.

2. Evaluating the function at the corner points of the region:
   Since we are looking at the boundary values, let's calculate $f(x, y)$ at the four corners of the rectangle given by the inequalities:

   • $(-3, -3)$
   • $(-3, 3)$
   • $(3, -3)$
   • $(3, 3)$

   Calculating these: $f(-3, -3) = 5(-3) + 2(-3) = -15 - 6 = -21$ $f(-3, 3) = 5(-3) + 2(3) = -15 + 6 = -9$ $f(3, -3) = 5(3) + 2(-3) = 15 - 6 = 9$ $f(3, 3) = 5(3) + 2(3) = 15 + 6 = 21$

3. Conclusion:

   • The absolute minimum value of $f(x, y)$ occurs at $(-3, -3)$, and it is $-21$.
   • The absolute maximum value of $f(x, y)$ occurs at $(3, 3)$, and it is $21$.

Thus, the absolute minimum is $-21$ and the absolute maximum is $21$.

Would you like further details on this solution or have any other questions?

Here are 5 related questions for more practice:

1. How do we find critical points for a non-linear function in two variables?
2. How would the solution change if the function had $x^2$ or $y^2$ terms?
3. Can a linear function have an absolute maximum or minimum inside the domain rather than on the boundary?
4. How does the shape of the region affect where the extrema are found?
5. What would happen if the boundary was a circle rather than a rectangle?

Tip: Always check the boundaries when solving optimization problems on a closed region!