Question

Consider the function \( f(x) = -2x^3 - 9x^2 - 12x + 5 \). Then, which of the following is/are correct?

• A. \( f(x) \) decreasing in \((0, \infty)\) and increasing in \((-\infty, 0)\)
• B. \( f(x) \) increasing in \((-\infty, 1) \cup (2, \infty)\) and decreasing in \((1, 2)\)
• C. \( f(x) \) increasing in \((0, \infty)\) and decreasing in \((-\infty, 0)\)
• D. \( f(x) \) is increasing in \((-2, -1)\) and decreasing in \((-\infty, -2) \cup (-1, \infty)\)

Hint:

To determine intervals of increasing/decreasing, the sign of the first derivative \( f'(x) \) is crucial. Always factor \( f'(x) \) to easily analyze its sign across intervals defined by its roots. Use test points within each interval to determine the sign.

Answer 1

The Correct Option is D