Let's solve part (a) of the question.

The profit function of the company is given by:

$P(x, y) = 1200 + 80x - 2x^2 + 100y - y^2$

where $x$ and $y$ represent the cost of a unit of labor and a unit of goods, respectively. We need to maximize this profit function by finding the critical points of $x$ and $y$, which involves calculating the first partial derivatives of $P(x, y)$ with respect to $x$ and $y$ and setting them equal to zero.

Step 1: First partial derivative with respect to $x$

$\frac{\partial P}{\partial x} = 80 - 4x$

Step 2: First partial derivative with respect to $y$

$\frac{\partial P}{\partial y} = 100 - 2y$

Step 3: Set the partial derivatives equal to zero to find the critical points

$80 - 4x = 0 \quad \Rightarrow \quad x = \frac{80}{4} = 20$

$100 - 2y = 0 \quad \Rightarrow \quad y = \frac{100}{2} = 50$

Thus, the values of $x$ and $y$ that maximize profit are $x = 20$ and $y = 50$.

Step 4: Verify that these are maximum points by checking the second derivatives (optional)

We can compute the second-order partial derivatives to ensure that these points indeed correspond to a maximum. However, the given function's structure (with negative coefficients for the quadratic terms in both $x$ and $y$) suggests it is a concave function, meaning the critical points correspond to a maximum.

Would you like to see the full second derivative verification?

Here are 5 questions to expand on this:

1. What is the significance of setting the first derivative equal to zero in optimization problems?
2. How do you interpret the result of $x = 20$ and $y = 50$ in a real business context?
3. How would you check if the critical points correspond to a maximum, minimum, or saddle point?
4. Can this method be applied to non-quadratic profit functions, and how?
5. How would constraints on labor or goods (if given) affect this optimization problem?

Tip: Always check for second-order conditions when optimizing to confirm whether a critical point is a maximum or minimum.