The interval on which the function `f(x)=-2x^(3)-9x^(2)-12x+1` is increasing is :

Options

A. `[-1,oo)`
B. [-2,-1]
C. `(-oo,-2]`
D. [-1,1]

Answer

To determine the interval on which the function \( f(x) = -2x^3 - 9x^2 - 12x + 1 \) is increasing, we will follow these steps:

### Step 1: Find the derivative of the function
To find the intervals where the function is increasing, we first need to compute the derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(-2x^3 - 9x^2 - 12x + 1) \]

Using the power rule for differentiation, we get:

\[ f'(x) = -6x^2 - 18x - 12 \]

### Step 2: Set the derivative greater than zero
The function \( f(x) \) is increasing where its derivative is positive. Thus, we set up the inequality:

\[ -6x^2 - 18x - 12 > 0 \]

### Step 3: Simplify the inequality
We can simplify this inequality by dividing through by -6 (remembering to reverse the inequality sign):

\[ x^2 + 3x + 2 < 0 \]

### Step 4: Factor the quadratic expression
Next, we factor the quadratic expression:

\[ x^2 + 3x + 2 = (x + 1)(x + 2) \]

Thus, we have:

\[ (x + 1)(x + 2) < 0 \]

### Step 5: Determine the critical points
The critical points from the factors are \( x = -1 \) and \( x = -2 \). These points divide the number line into intervals. We will test the sign of the product in each interval:

1. \( (-\infty, -2) \)
2. \( (-2, -1) \)
3. \( (-1, \infty) \)

### Step 6: Test the intervals
- For \( x < -2 \) (e.g., \( x = -3 \)):
\[ (-3 + 1)(-3 + 2) = (-2)(-1) = 2 > 0 \]

- For \( -2 < x < -1 \) (e.g., \( x = -1.5 \)):
\[ (-1.5 + 1)(-1.5 + 2) = (-0.5)(0.5) = -0.25 < 0 \]

- For \( x > -1 \) (e.g., \( x = 0 \)):
\[ (0 + 1)(0 + 2) = (1)(2) = 2 > 0 \]

### Step 7: Conclusion
The function \( f(x) \) is increasing in the interval where the product is negative, which is:

\[ (-2, -1) \]

### Final Answer
The interval on which the function \( f(x) \) is increasing is \( (-2, -1) \).

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