The function given is:

$z = f(x, y) = (x - 4)^2 + (y - 2)^2 + 46$

We are asked to find the boundary points for the constraints $0 \leq x \leq 15$ and $0 \leq y \leq 13$.

Step 1: Identify the Boundary Points

The boundary points occur at the corners of the region defined by the constraints. These corners are:

• $(x, y) = (0, 0)$
• $(x, y) = (0, 13)$
• $(x, y) = (15, 0)$
• $(x, y) = (15, 13)$

Step 2: Calculate $z$ at Each Boundary Point

For each corner point, substitute the values of $x$ and $y$ into the function $z = (x - 4)^2 + (y - 2)^2 + 46$.

1. At $(x, y) = (0, 0)$: $z = (0 - 4)^2 + (0 - 2)^2 + 46 = 16 + 4 + 46 = 66$ The point is $(0, 0, 66)$.

2. At $(x, y) = (0, 13)$: $z = (0 - 4)^2 + (13 - 2)^2 + 46 = 16 + 121 + 46 = 183$ The point is $(0, 13, 183)$.

3. At $(x, y) = (15, 0)$: $z = (15 - 4)^2 + (0 - 2)^2 + 46 = 121 + 4 + 46 = 171$ The point is $(15, 0, 171)$.

4. At $(x, y) = (15, 13)$: $z = (15 - 4)^2 + (13 - 2)^2 + 46 = 121 + 121 + 46 = 288$ The point is $(15, 13, 288)$.

Step 3: List the Boundary Points

The four boundary points in $(x, y, z)$ form are:

1. $(0, 0, 66)$
2. $(0, 13, 183)$
3. $(15, 0, 171)$
4. $(15, 13, 288)$

Let me know if you'd like more details or have any questions!

Here are 5 related questions to explore:

1. How do we determine if these boundary points include global minima or maxima?
2. What is the role of the constant $46$ in the function $z$?
3. Can you explain how this problem would change if the constraints were different?
4. How can Lagrange multipliers be used if we had equality constraints?
5. What would happen to the boundary points if the domain of $x$ and $y$ was expanded?

Tip: Always double-check your constraint boundaries to ensure the correct points are calculated, especially when solving optimization problems!