\(\mathrm{f(x) = 120 - (x - 10)^2}\) What is the maximum value of the function f?...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = 120 - (x - 10)^2}\)
• Find: Maximum value of function f

2. INFER the solution strategy

• To maximize \(\mathrm{f(x)}\), we need to make \(\mathrm{120 - (x - 10)^2}\) as large as possible
• Since we're subtracting \(\mathrm{(x - 10)^2}\) from 120, we need to subtract the smallest possible amount
• This means we need to minimize \(\mathrm{(x - 10)^2}\)

3. INFER the minimum value of the squared term

• The expression \(\mathrm{(x - 10)^2}\) is a perfect square
• Any real number squared gives a non-negative result: \(\mathrm{(x - 10)^2 \geq 0}\)
• The smallest possible value is 0

4. INFER when the minimum occurs

• \(\mathrm{(x - 10)^2 = 0}\) when \(\mathrm{x - 10 = 0}\)
• This happens when \(\mathrm{x = 10}\)

5. SIMPLIFY to find the maximum value

• When \(\mathrm{x = 10}\): \(\mathrm{f(10) = 120 - (10 - 10)^2}\)
• \(\mathrm{f(10) = 120 - 0^2 = 120 - 0 = 120}\)

Answer: D) 120

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the optimization strategy. They might try to find where the derivative equals zero or attempt to plug in specific values without understanding that minimizing the squared term maximizes the function. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about function behavior: Students might think that since \(\mathrm{(x-10)^2}\) can get very large, the function can somehow exceed 120. They may not grasp that subtracting larger amounts from 120 makes the result smaller, not larger. This may lead them to select Choice C (110) by incorrectly reasoning about the function's behavior.

The Bottom Line:

This problem requires recognizing that optimization of \(\mathrm{f(x) = 120 - (squared\ term)}\) depends on minimizing the squared term. The key insight is understanding how subtraction affects maximization: to make the whole expression larger, make what you're subtracting smaller.