Which of the following functions has(have) a minimum value at -{3}?\(\mathrm{f(x) = -6(3)^x - 3}\)\(\mathrm{g(x) = -3(6)^x}\)

1. INFER the structure of each function

• Function I: \(\mathrm{f(x) = -6(3)^x - 3}\)
  • This is exponential form: \(\mathrm{f(x) = a(b)^x + c}\) where \(\mathrm{a = -6, b = 3, c = -3}\)

• Function II: \(\mathrm{g(x) = -3(6)^x}\)
  • This is exponential form: \(\mathrm{f(x) = a(b)^x}\) where \(\mathrm{a = -3, b = 6}\)

2. INFER the behavior of each function

• Both functions have \(\mathrm{a \lt 0}\) and \(\mathrm{b \gt 1}\)
• For exponential functions, when \(\mathrm{a \lt 0}\) and \(\mathrm{b \gt 1}\), the function is decreasing
• As x increases: \(\mathrm{b^x}\) increases → \(\mathrm{a(b^x)}\) becomes more negative → function value decreases

3. INFER whether minimum values exist

• Since both functions are decreasing, they continue to get smaller as x increases
• Decreasing functions don't have minimum values - they approach negative infinity
• Therefore, neither function has a minimum value

Answer: D. Neither I nor II

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize how the signs and values of parameters determine exponential function behavior. They might think that because the functions involve exponentials, they automatically have minimum or maximum values somewhere.

This often leads to confusion about which functions are increasing vs decreasing, potentially causing them to guess or select Choice C (I and II) thinking both functions must have extrema.

Second Most Common Error:

Conceptual confusion about minimum values: Students may misinterpret "minimum value at -3" as asking for the function value when \(\mathrm{x = -3}\), rather than asking whether the function has a minimum value at all.

This misunderstanding of what constitutes a "minimum value" may lead them to calculate \(\mathrm{f(-3)}\) or \(\mathrm{g(-3)}\) and select Choice A (I only) or Choice B (II only) based on which calculation seems more reasonable.

The Bottom Line:

This problem requires students to analyze exponential function behavior systematically rather than plugging in values. The key insight is recognizing that parameter analysis (specifically \(\mathrm{a \lt 0, b \gt 1}\)) immediately tells us these functions are decreasing and therefore cannot have minimum values.