Question

Consider the function given below and pick one or more CORRECT statement(s) from the following choices.
\[ f(x) = x^3 - \frac{15}{2} x^2 + 18x + 20 \]

• A. \( f(x) \) has a local minimum at \( x = 3 \)
• B. \( f(x) \) has a local maximum at \( x = 3 \)
• C. \( f(x) \) has a local minimum at \( x = 2 \)
• D. \( f(x) \) has a local maximum at \( x = 2 \)

Hint:

Find f'(x) and set it to zero for critical points. Use f''(x) to classify them as maxima or minima. Analyze end behavior and check where f(x) = 0 for root insights.

Answer 1

The Correct Option is A, D

Given the function:

\[ f(x) = x^3 - \frac{15}{2} x^2 + 18x + 20 \]

Step 1: Compute the first derivative:

\[ f'(x) = 3x^2 - 15x + 18 \]

Setting \( f'(x) = 0 \) to find the critical points:

\[ 3x^2 - 15x + 6 = 0 \] \[ x^2 - 5x + 2 = 0 \]

Solving for \( x \):

\[ x = 2, \quad x = 3 \]

Step 2: Compute the second derivative:

\[ f''(x) = 6x - 15 \]

Evaluating at \( x = 2 \):

\[ f''(2) = 6(2) - 15 = -3 \quad \text{(Local Maximum)} \]

Evaluating at \( x = 3 \):

\[ f''(3) = 6(3) - 15 = 3 \quad \text{(Local Minimum)} \]

Step 3: Compute function values at critical points:

\[ f(2) = (2)^3 - \frac{15}{2} (2)^2 + 18(2) + 20 = 34 \]

\[ f(3) = (3)^3 - \frac{15}{2} (3)^2 + 18(3) + 20 = 33.5 \]

Conclusion:

The local maximum occurs at \( x = 2 \) and the local minimum occurs at \( x = 3 \). Therefore, the correct answers are (A) and (D).