From the provided image, the goal is to find $y^*$ where $\frac{\partial \Pi}{\partial y} = 0$. Here's the breakdown of the steps to solve for $y^*$:

1. Profit Function: $\Pi = py - \left[ (w)(l) + (r)(k) \right]$

2. Derivative of Profit with respect to $y$: $\frac{\partial \Pi}{\partial y} = p - \frac{\partial}{\partial y} \left[ (w)(l) + (r)(k) \right]$

   From the equation in the image: $\frac{\partial \Pi}{\partial y} = p - \left[ 20l + 4k \right]$

   This derivative is set to zero to maximize profits.

3. Set $\frac{\partial \Pi}{\partial y} = 0$: $p - \left( 20l + 4k \right) = 0$

4. Solve for $y^*$: To proceed, you'll need to determine the relationship between $y$, $l$, and $k$, which might involve production functions or constraints not shown here. If these are provided in your full notes or problem, substitute those relationships and solve for $y$.

If you share more details or constraints from the problem, I can help you fully compute $y^*$!

Questions for Further Understanding:

1. What is the production function $y = f(l, k)$ used here, if any?
2. Are $l$ and $k$ given as functions of $y$, or are they constants?
3. What is the value of $p$ in this scenario?
4. Does the problem involve specific cost functions or constraints for $l$ and $k$?
5. Is this part of a larger optimization problem involving multiple variables?

Tip:

When maximizing profit, always ensure that the marginal cost (MC) and marginal revenue (MR) are aligned correctly. Double-check the production function dependencies when solving.