The problem provides the function $f(x, y) = x^2 + y^2 + 2xy + 5$ and asks us to find the absolute maximum and minimum on the region $D = \{(x, y) : |x| \leq 1, |y| \leq 1\}$, which defines a square region with corners at $(-1, -1), (-1, 1), (1, -1), (1, 1)$.

1. Steps to find the absolute maximum and minimum:

Step 1: Find the partial derivatives.

We need the critical points by setting the partial derivatives of $f(x, y)$ equal to zero.

• Partial derivative with respect to $x$: $f_x = 2x + 2y$
• Partial derivative with respect to $y$: $f_y = 2y + 2x$

Setting both derivatives to 0: $f_x = 2x + 2y = 0 \quad \text{and} \quad f_y = 2y + 2x = 0$

Both equations are identical, so they reduce to: $x + y = 0 \quad \Longrightarrow \quad y = -x$

Step 2: Check the critical point.

Using $y = -x$ and substituting it back into the region $|x| \leq 1, |y| \leq 1$, we test the origin $(x, y) = (0, 0)$.

Evaluating the function at this point: $f(0, 0) = 0^2 + 0^2 + 2(0)(0) + 5 = 5$

Thus, the function takes the value 5 at the origin.

Step 3: Evaluate the function on the boundaries of $D$.

We now check the boundary of the region, which consists of the lines $x = \pm 1$ and $y = \pm 1$.

• On $x = 1$: $f(1, y) = 1^2 + y^2 + 2(1)(y) + 5 = y^2 + 2y + 6$ This simplifies to $(y + 1)^2 + 5$. The minimum occurs at $y = -1$, giving: $f(1, -1) = (-1 + 1)^2 + 5 = 5$ The maximum occurs at $y = 1$, giving: $f(1, 1) = (1 + 1)^2 + 5 = 9$

• On $x = -1$: $f(-1, y) = 1 + y^2 - 2y + 5 = (y - 1)^2 + 5$ The minimum occurs at $y = 1$, giving: $f(-1, 1) = (1 - 1)^2 + 5 = 5$ The maximum occurs at $y = -1$, giving: $f(-1, -1) = (-1 - 1)^2 + 5 = 9$

• On $y = 1$ or $y = -1$, similar calculations yield the same results:

  • Minimum value: $5$ at points $(1, -1)$ and $(-1, 1)$
  • Maximum value: $9$ at points $(1, 1)$ and $(-1, -1)$

Step 4: Conclusion.

• Absolute Maximum: $f(x, y) = 9$ at $(1, 1)$ and $(-1, -1)$.
• Absolute Minimum: $f(x, y) = 5$ at the origin $(0, 0)$, as well as at $(1, -1)$ and $(-1, 1)$.

Final Answers:

1. Absolute Maximum: $9$
2. Absolute Minimum: $5$

Would you like further details or a visualization of the region?

Here are 5 related questions to deepen your understanding:

1. How would the solution change if the region $D$ were a circle instead of a square?
2. What happens if the function is modified to $f(x, y) = x^2 + y^2 - 2xy + 5$?
3. How do we determine if a critical point is a minimum or maximum using the second derivative test?
4. What role do boundaries play in constrained optimization problems?
5. Can the method of Lagrange multipliers be applied here to find extrema?

Tip: In optimization problems, always consider both critical points and boundaries when working on closed regions.