Question

The function \( f(x) = 2x^3 - 3x^2 - 12x + 4, \, x \in \mathbb{R \) has:}

• A. two points of local maximum.
• B. two points of local minimum.
• C. one local maximum and one local minimum.
• D. neither maximum nor minimum.

Hint:

For cubic polynomials: \begin{itemize} \item Usually two critical points. \item Sign of second derivative determines nature. \end{itemize}

Answer 1

The Correct Option is C

Concept:
To determine local extrema:

• Find critical points using \( f'(x) = 0 \)
• Use the second derivative test

Step 1: Find first derivative.
\[ f'(x) = 6x^2 - 6x - 12 \] \[ = 6(x^2 - x - 2) \] \[ = 6(x - 2)(x + 1) \] Critical points: \[ x = 2, \quad x = -1 \]
Step 2: Second derivative test.
\[ f''(x) = 12x - 6 \] At \( x = 2 \): \[ f''(2) = 24 - 6 = 18 > 0 \] So local minimum.
At \( x = -1 \): \[ f''(-1) = -12 - 6 = -18 < 0 \] So local maximum.

Step 3: Conclusion.
One local maximum and one local minimum.