The task involves completing a graph that illustrates the relationships between revenue $E(x)$, profit $G(x)$, and costs $K(x)$, where:

1. $E(x)$ is the revenue as a function of the quantity $x$ sold.
2. $G(x)$ is the profit, defined as $G(x) = E(x) - K(x)$, where $K(x)$ is the cost function.
3. $K(x)$ is assumed to be linear, as stated in the problem.

From the provided graph:

• $E(x)$ appears as a concave curve (likely quadratic), indicating that revenue increases at a decreasing rate as $x$ grows.
• $G(x)$ is shown as a curve below $E(x)$, with $G(x)$ initially positive but eventually decreasing and crossing zero.
• The task is to draw the linear cost function $K(x)$, which satisfies $G(x) = E(x) - K(x)$.

Key Observations:

1. $K(x)$ should be a straight line intersecting $E(x)$ at the point where $G(x) = 0$ (the break-even points).
2. Since $K(x)$ is linear, it has a constant slope.

Would you like me to provide a step-by-step method to calculate or sketch $K(x)$, or clarify any specific part of the graph or functions? Let me know how I can assist further!

Relative Questions:

1. How is the cost function $K(x)$ typically calculated if its linearity is assumed?
2. Why does $G(x)$ cross the $x$-axis, and what does this represent in terms of profit?
3. What is the significance of $E(x)$ being a concave function in practical scenarios?
4. How can we mathematically determine the slope of $K(x)$ based on $G(x)$ and $E(x)$?
5. What economic assumptions underlie the idea that all produced quantities are sold?

Tip:

Always check the intersection points between $E(x)$ and $K(x)$, as they provide critical insights into profitability and cost-effectiveness.