For the function g defined by \(\mathrm{g(x) = 5 - |x + 3|}\), what is the maximum value?

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = 5 - |x + 3|}\)
• Find: Maximum value of \(\mathrm{g(x)}\)

2. INFER the optimization strategy

• To maximize \(\mathrm{g(x) = 5 - |x + 3|}\), we need to minimize \(\mathrm{|x + 3|}\)
• Since we're subtracting \(\mathrm{|x + 3|}\) from 5, the smaller \(\mathrm{|x + 3|}\) becomes, the larger \(\mathrm{g(x)}\) becomes

3. INFER the minimum value of the absolute value

• By definition, \(\mathrm{|x + 3| \geq 0}\) for all real numbers \(\mathrm{x}\)
• The minimum possible value of \(\mathrm{|x + 3|}\) is 0
• This minimum occurs when \(\mathrm{x + 3 = 0}\), which means \(\mathrm{x = -3}\)

4. SIMPLIFY to find the maximum

• When \(\mathrm{x = -3}\):
  \(\mathrm{g(-3) = 5 - |(-3) + 3|}\)
  \(\mathrm{= 5 - |0|}\)
  \(\mathrm{= 5 - 0}\)
  \(\mathrm{= 5}\)
• Therefore, the maximum value of \(\mathrm{g(x)}\) is 5

Answer: D (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the optimization relationship between maximizing \(\mathrm{g(x)}\) and minimizing \(\mathrm{|x + 3|}\)

Instead, they might try to find where the derivative equals zero (treating it like a standard calculus problem) or randomly substitute values without understanding the underlying strategy. This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about absolute value: Students might think \(\mathrm{|x + 3|}\) can be negative or forget that absolute value expressions have a minimum value of 0

This misconception could lead them to incorrectly calculate values, potentially selecting Choice A (-3) if they confuse the x-value where the maximum occurs with the maximum value itself.

The Bottom Line:

Success requires recognizing that maximizing a function of the form "constant minus absolute value" means minimizing the absolute value term. The key insight is understanding when absolute value expressions reach their minimum.