Question

question 2 (6 points)

let \\(f(x) = x^3 - 6x^2 + 9x + 1\\).

select all true statements.

\\(f\\) has a local maximum at \\(x = 1\\)
\\(f\\) is decreasing on \\((1, 3)\\)
\\(f\\) has a local maximum at \\(x = 3\\)
\\(f\\) has a local minimum at \\(x = 3\\)
\\(f\\) is increasing on \\((-\infty, 1)\\) and \\((3, \infty)\\)

Explanation:

Find the derivative of the function

Using the Derivative of Polynomials knowledge point

$$ f'(x) = 3x^2 - 12x + 9 $$

Find the critical points

Using the Critical Points knowledge point

$$
LATEXBLOCK0
$$

Determine increasing and decreasing intervals

Using the Increasing and Decreasing Intervals knowledge point

$$
LATEXBLOCK1
$$

Classify local extrema

Using the Local Extrema knowledge point

$$
LATEXBLOCK2
$$

Evaluate the statements

We compare our findings with the given options:

• "\(f\) has a local maximum at \(x = 1\)": True.
• "\(f\) is decreasing on \((1, 3)\)": True.
• "\(f\) has a local maximum at \(x = 3\)": False (it is a local minimum).
• "\(f\) has a local minimum at \(x = 3\)": True.
• "\(f\) is increasing on \((-\infty, 1)\) and \((3, \infty)\)": True.

Answer:

• f has a local maximum at \(x = 1\) (Correct answer)
• f is decreasing on \((1, 3)\) (Correct answer)
• f has a local maximum at \(x = 3\)
• f has a local minimum at \(x = 3\) (Correct answer)
• f is increasing on \((-\infty, 1)\) and \((3, \infty)\) (Correct answer)