A small business models its daily revenue R, in hundreds of dollars, as a function of the selling price p,...

1. INFER the problem strategy

• Given: \(\mathrm{R(p) = (p - 2)(12 - p)}\) represents revenue as a quadratic function
• Need: The price p that maximizes revenue
• Key insight: Since this is a downward-opening parabola, the maximum occurs at the vertex

2. INFER which method to use for finding the vertex

• Since the function is in factored form, we can use the root-midpoint method
• Alternative: Expand to standard form and use the vertex formula
• Both approaches will give the same result

3. SIMPLIFY using the root-midpoint method

• Find the roots by setting each factor to zero:
  • \(\mathrm{p - 2 = 0 \rightarrow p = 2}\)
  • \(\mathrm{12 - p = 0 \rightarrow p = 12}\)
• The vertex occurs at the midpoint: \(\mathrm{p = (2 + 12)/2 = 14/2 = 7}\)

4. APPLY CONSTRAINTS to select the correct interval

• We found \(\mathrm{p = 7}\)
• Check which interval contains 7:
  • (A) \(\mathrm{0 \lt p \lt 2}\): No, \(\mathrm{7 \gt 2}\)
  • (B) \(\mathrm{2 \lt p \lt 4}\): No, \(\mathrm{7 \gt 4}\)
  • (C) \(\mathrm{4 \lt p \lt 6}\): No, \(\mathrm{7 \gt 6}\)
  • (D) \(\mathrm{6 \lt p \lt 8}\): Yes, \(\mathrm{6 \lt 7 \lt 8}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect that "maximizing a quadratic function" means "finding the vertex of a parabola." They might try to solve \(\mathrm{R(p) = 0}\) or look for other patterns, missing the fundamental relationship between optimization and vertex location. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand they need the vertex but make arithmetic errors. Common mistakes include calculating \(\mathrm{(2 + 12)/2 = 6}\) instead of 7, or if using the vertex formula, errors in \(\mathrm{-14/(-2)}\). These errors typically lead them to select Choice (C) (\(\mathrm{4 \lt p \lt 6}\)) if they get \(\mathrm{p = 6}\), or other incorrect intervals based on their miscalculation.

The Bottom Line:

This problem tests whether students can connect the abstract concept of "maximizing a function" to the concrete geometric concept of "finding a parabola's vertex." The calculation itself is straightforward once the connection is made, but many students get stuck at the conceptual bridge between optimization and vertex-finding.