Question

The function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is decreasing in the interval:

• A. \( [-\infty, \infty] \)
• B. \( (-2, -1) \)
• C. \( (-\infty, -2] \)
• D. \( [-1, 0] \)
• E. \( (-1, 1] \)

Hint:

A function is decreasing when its derivative is negative, so always check the sign of \( f'(x) \) in the intervals between critical points.

Answer 1

The Correct Option is B

To find where the function is decreasing, we need to find the critical points by first calculating the derivative of \( f(x) \): \[ f'(x) = 6x^2 + 18x + 12 \] Now, solve for \( f'(x) = 0 \) to find the critical points: \[ 6x^2 + 18x + 12 = 0 \] \[ x^2 + 3x + 2 = 0 \] \[ (x + 1)(x + 2) = 0 \] So, \( x = -1 \) and \( x = -2 \). To determine where the function is decreasing, evaluate the sign of \( f'(x) \) in the intervals \( (-\infty, -2) \), \( (-2, -1) \), and \( (-1, \infty) \). For \( x \in (-2, -1) \), the function is decreasing.