Because the coefficient of $x^2$ is negative, does the parabola open upward or downward? For that shape, where does the maximum value occur on the graph?

Write $R(x)$ in the form $ax^2 + bx + c$ and identify $a$ and $b$. Then, recall and use the formula for the $x$-coordinate of the vertex, $x = -\frac{b}{2a}$.

In Desmos, enter the function exactly as given: R(x) = -5x^2 + 200x. This will display a downward-opening parabola.

Tap or click on the parabola; Desmos will offer to show the maximum point. Select it and note the x-value of this maximum point—that $x$ tells you how many smartwatches give the maximum revenue.

You can also tap where the graph crosses the x-axis to see the two x-intercepts. The maximum occurs halfway between these two x-values. Find the midpoint of those intercepts to get the $x$-value that maximizes revenue.

The revenue is modeled by the quadratic function

This is a parabola opening downward because the coefficient of $x^2$ is negative. For a downward-opening parabola, the maximum value occurs at the vertex. The question is asking for the $x$-value (number of smartwatches) at that maximum point.

Write the quadratic in standard form $ax^2 + bx + c$:

So:

• $a = -5$
• $b = 200$
• $c = 0$

For any quadratic $y = ax^2 + bx + c$, the $x$-coordinate of the vertex is given by

We will use this formula to find the number of smartwatches that gives the maximum revenue.

Substitute $a = -5$ and $b = 200$ into $x = -\frac{b}{2a}$:

So, according to the model, the company must produce and sell 20 smartwatches per day to achieve the maximum possible revenue. This corresponds to answer choice B) 20.