A point P lies on the x-axis, so its coordinates can be written as \(\mathrm{(t, 0)}\) for some real number...

1. TRANSLATE the problem information

• Given information:
  • Point P lies on x-axis, so \(\mathrm{P = (t, 0)}\) for some real number \(\mathrm{t}\)
  • Fixed point is \(\mathrm{(3, 4)}\)
  • Need to find minimum distance from P to \(\mathrm{(3, 4)}\)

2. TRANSLATE using distance formula

• Distance from \(\mathrm{P = (t, 0)}\) to \(\mathrm{(3, 4)}\):
  \(\mathrm{d = \sqrt{(t-3)^2 + (0-4)^2}}\)
  \(\mathrm{d = \sqrt{(t-3)^2 + 16}}\)

3. INFER the minimization strategy

• To minimize \(\mathrm{d}\), we need to minimize the expression under the square root
• Since square root is an increasing function, minimizing \(\mathrm{d}\) is equivalent to minimizing \(\mathrm{(t-3)^2 + 16}\)
• The constant 16 doesn't change, so we need to minimize \(\mathrm{(t-3)^2}\)

4. INFER when the minimum occurs

• Since \(\mathrm{(t-3)^2 \geq 0}\) for all real numbers \(\mathrm{t}\)
• The minimum value of \(\mathrm{(t-3)^2}\) is 0, which happens when \(\mathrm{t-3 = 0}\)
• Therefore: \(\mathrm{t = 3}\)

5. SIMPLIFY to find the minimum distance

• When \(\mathrm{t = 3}\):
  \(\mathrm{d = \sqrt{(3-3)^2 + 16}}\)
  \(\mathrm{d = \sqrt{0 + 16}}\)
  \(\mathrm{d = \sqrt{16}}\)
  \(\mathrm{d = 4}\)

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that minimizing the distance requires minimizing the expression under the square root. Students might try complicated calculus approaches or get confused about what to optimize.

This leads to confusion and abandoning systematic solution, causing them to guess among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making arithmetic errors when evaluating \(\mathrm{\sqrt{16}}\), perhaps confusing it with other perfect squares or making calculation mistakes.

This may lead them to select Choice A (2) or Choice B (3) instead of the correct answer.

The Bottom Line:

This problem tests whether students can recognize that optimization problems involving distances often reduce to minimizing quadratic expressions, and that they understand when such expressions achieve their minimum values.