In the xy-plane, consider the function \(\mathrm{F(x, y) = (x - 2)^2 + (y + 5)^2 + 7}\). The variables...

1. INFER the structure of the function

• Given: \(\mathrm{F(x, y) = (x - 2)^2 + (y + 5)^2 + 7}\)
• Key insight: This function consists of two squared terms plus a constant
• Since squares are always non-negative, each squared term has a minimum value of 0

2. INFER the minimization strategy

• To minimize F(x, y), we need to minimize each squared term separately
• The minimum value of \(\mathrm{(x - 2)^2}\) is 0, occurring when \(\mathrm{x - 2 = 0}\)
• The minimum value of \(\mathrm{(y + 5)^2}\) is 0, occurring when \(\mathrm{y + 5 = 0}\)
• Both conditions can be satisfied simultaneously

3. SIMPLIFY to find the critical values

• From \(\mathrm{x - 2 = 0}\): \(\mathrm{x = 2}\)
• From \(\mathrm{y + 5 = 0}\): \(\mathrm{y = -5}\)

4. SIMPLIFY to evaluate the minimum

• At \(\mathrm{x = 2}\), \(\mathrm{y = -5}\):
  \(\mathrm{F(2, -5) = (2 - 2)^2 + (-5 + 5)^2 + 7}\)
  \(\mathrm{= 0 + 0 + 7}\)
  \(\mathrm{= 7}\)

Answer: B (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that the function structure allows for simultaneous minimization of both squared terms. They might try calculus approaches or get confused about when both squared terms equal zero at the same time. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that squared terms should equal zero but make algebraic errors when solving \(\mathrm{x - 2 = 0}\) or \(\mathrm{y + 5 = 0}\), particularly with the negative sign in \(\mathrm{(y + 5)^2}\). For example, thinking \(\mathrm{y + 5 = 0}\) gives \(\mathrm{y = 5}\) instead of \(\mathrm{y = -5}\). This may lead them to evaluate the function at incorrect coordinates and select a wrong answer.

The Bottom Line:

This problem tests whether students can recognize the optimization structure of functions with squared terms and understand that the minimum occurs when all squared components simultaneously equal zero.