Which of the following functions has(have) a maximum value at x = 4? \(\mathrm{f(x) = -\sqrt{x-1} + 3}\) \(\mathrm{g(x) =...

1. TRANSLATE the problem information

• Given information:
  • Function I: \(\mathrm{f(x) = -\sqrt{x-1} + 3}\)
  • Function II: \(\mathrm{g(x) = \sqrt{x+2} - 5}\)
  • Need to determine which has a maximum value at \(\mathrm{x = 4}\)
• What this means: We need to determine if \(\mathrm{x = 4}\) is a local maximum point for either function

2. INFER the approach for each function

• Strategy: Analyze the behavior (increasing/decreasing) of each function
• Key insight: If a function is strictly increasing or decreasing, it cannot have a local maximum at interior points of its domain

3. Analyze Function I: \(\mathrm{f(x) = -\sqrt{x-1} + 3}\)

• INFER the domain: Since we need \(\mathrm{x - 1 \geq 0}\), the domain is \(\mathrm{x \geq 1}\)
• INFER the behavior:
  • As x increases from 1, the term \(\mathrm{\sqrt{x-1}}\) increases
  • Therefore \(\mathrm{-\sqrt{x-1}}\) decreases as x increases
  • So \(\mathrm{f(x) = -\sqrt{x-1} + 3}\) is strictly decreasing for \(\mathrm{x \gt 1}\)
• INFER the conclusion: Since \(\mathrm{f(x)}\) is decreasing, it cannot have a maximum at \(\mathrm{x = 4}\)

4. Analyze Function II: \(\mathrm{g(x) = \sqrt{x+2} - 5}\)

• INFER the domain: Since we need \(\mathrm{x + 2 \geq 0}\), the domain is \(\mathrm{x \geq -2}\)
• INFER the behavior:
  • As x increases from -2, the term \(\mathrm{\sqrt{x+2}}\) increases
  • Therefore \(\mathrm{g(x) = \sqrt{x+2} - 5}\) is strictly increasing for \(\mathrm{x \gt -2}\)
• INFER the conclusion: Since \(\mathrm{g(x)}\) is increasing, it cannot have a maximum at \(\mathrm{x = 4}\)

5. INFER the final answer

• Neither function has a maximum at \(\mathrm{x = 4}\)
• Both functions are monotonic (one decreasing, one increasing)

Answer: D) Neither I nor II

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on evaluating \(\mathrm{f(4)}\) and \(\mathrm{g(4)}\) instead of analyzing function behavior

Students calculate \(\mathrm{f(4) = 3 - \sqrt{3} \approx 1.27}\) and \(\mathrm{g(4) = \sqrt{6} - 5 \approx -2.55}\), then assume that because these are specific values, one or both must represent maxima. They miss that finding a maximum requires comparing the function value at \(\mathrm{x = 4}\) to nearby values, which means understanding whether the function is increasing or decreasing around that point.

This may lead them to select Choice A (I only) if they think the larger value indicates a maximum, or Choice C (I and II) if they think both calculated values represent maxima.

Second Most Common Error:

Poor INFER reasoning about function transformations: Students incorrectly analyze the behavior of \(\mathrm{f(x) = -\sqrt{x-1} + 3}\)

Students see the square root and assume the function is increasing, forgetting that the negative sign in front of \(\mathrm{\sqrt{x-1}}\) reverses the behavior. They might think: "Square root functions increase, so this increases too," missing that \(\mathrm{-\sqrt{x-1}}\) actually decreases as x increases.

This leads to confusion about which functions could have maxima, potentially causing them to select Choice B (II only) or get stuck and guess.

The Bottom Line:

This problem requires students to move beyond just plugging in \(\mathrm{x = 4}\) and instead analyze the overall behavior of transformed functions. The key insight is recognizing that monotonic functions cannot have local extrema at interior points.