$f(x)=-2x^{3}-9x^{2}-12x+1$ is strictly decreasing in:

Question

Accepted Answer

To determine where the function $ f(x) = -2x^3 - 9x^2 - 12x + 1 $ is strictly decreasing, we need to analyze the derivative of the function, $ f'(x) $, and identify the intervals where it is negative. 1. **Find the derivative $ f'(x) $:** $$ f'(x) = \frac{d}{dx}(-2x^3 - 9x^2 - 12x + 1) = -6x^2 - 18x - 12 $$ 2. **Determine where $ f'(x) < 0 $:** We need to solve the inequality: $$ -6x^2 - 18x - 12 < 0 $$ Simplifying by dividing the entire inequality by -6 (and reversing the inequality sign): $$ x^2 + 3x + 2 > 0 $$ 3. **Factor the quadratic expression:** $$ x^2 + 3x + 2 = (x + 1)(x + 2) $$ 4. **Find the roots of the equation $ (x + 1)(x + 2) = 0 $:** The roots are $ x = -1 $ and $ x = -2 $. 5. **Determine the sign of $ (x + 1)(x + 2) $ in each interval:** - For $ x < -2 $, both $ x + 1 $ and $ x + 2 $ are negative, so the product is positive. - For $ -2 < x < -1 $, $ x + 1 $ is negative and $ x + 2 $ is positive, so the product is negative. - For $ x > -1 $, both $ x + 1 $ and $ x + 2 $ are positive, so the product is positive. 6. **Identify the intervals where $ f'(x) < 0 $:** From the analysis, $ f'(x) < 0 $ in the interval $ (-2, -1) $. 7. **Combine the intervals where $ f(x) $ is strictly decreasing:** The function $ f(x) $ is strictly decreasing where $ f'(x) < 0 $, which is in the intervals $ (-\infty, -2) $ and $ (-1, \infty) $. Thus, the function is strictly decreasing in the intervals $ (-\infty, -2) \cup (-1, \infty) $, which corresponds to Option 4. **Explanation of Incorrect Options:** - **Option 1: (-2, $\infty$)** - This option suggests that the function is decreasing from $-2$ to $\infty$, but the function is actually increasing in the interval $(-2, -1)$. - **Option 2: (-2, -1)** - This option only covers part of the interval where the function is decreasing. It misses the interval $(-\infty, -2)$ and $(-1, \infty)$. - **Option 3: (-$\infty$, -1)** - This option includes the interval $(-2, -1)$ where the function is increasing, thus it is incorrect. Therefore, Option 4 is the correct answer as it accurately describes the intervals where the function is strictly decreasing.