To solve this problem with a graphing calculator, here are the steps you can follow. The given function is:

$f(x) = -0.005x^2 + 120x - 100,000$

This function represents the profit based on the number of items $x$ produced and sold. You are asked to find the number of items $x$ that maximizes profit, which corresponds to finding the vertex of this quadratic function.

Steps on a Graphing Calculator (TI-84 as an example):

1. Input the Function:

   • Turn on the graphing calculator.
   • Press the Y= button.
   • Enter the equation: $-0.005x^2 + 120x - 100000$ in the form of Y1=.

2. Set the Window:

   • Press the WINDOW button to adjust the viewing window. Based on the context (the number of items should be positive), you can set:
     • Xmin = 0 (since the company can't sell negative items).
     • Xmax = 30,000 (to ensure the graph is visible within a reasonable range of items).
     • Ymin = -200,000 (to give space for possible negative profits).
     • Ymax = 200,000 (to allow a view of the positive profit area).

3. Graph the Function:

   • Press the GRAPH button to display the function.

4. Find the Maximum:

   • Since this is a quadratic equation with a downward-facing parabola (because of the negative coefficient of $x^2$), the maximum profit will occur at the vertex.
   • Press the 2nd button, then TRACE to access the CALC menu.
   • Select option 4: Maximum.
   • Move the cursor using the left and right arrow keys to the left of the peak of the graph and press ENTER.
   • Move the cursor to the right of the peak and press ENTER again.
   • The calculator will ask for a guess—move the cursor close to the peak and press ENTER.
   • The calculator will display the $x$ value of the maximum, which is the number of items that maximizes the profit, and the $y$ value, which represents the maximum profit.

5. Interpret the Result:

   • The $x$-value given is the number of items the company should produce and sell to maximize their profit.
   • Based on the function provided, this value should fall within one of the multiple-choice intervals (A-E) on the problem.

Would you like further assistance with interpreting the result?

Relative questions:

1. What is the significance of the vertex in a quadratic function related to profit maximization?
2. How can you determine whether a quadratic function has a maximum or minimum?
3. Why is it important to set a proper window for graphing in this scenario?
4. What does the negative coefficient of $x^2$ indicate about the profit function's graph?
5. How can you use the graph to estimate the profit for a different number of items?

Tip: Always check your graphing window to ensure that both $x$ and $y$ values make sense in the context of the problem, especially for real-world applications like profit analysis.