24. Find the intervals in which the following function is (a) increasing and (b) decreasing `f(x)=2x^3+9x^2+12x-1`

Options

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

Explanation

To find the intervals in which the function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is increasing or decreasing, we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \).

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

Using the power rule, we differentiate each term:

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

### Step 2: Set the derivative equal to zero
Next, we find the critical points by setting the derivative equal to zero:

\[ 6x^2 + 18x + 12 = 0 \]

### Step 3: Simplify the equation
We can simplify this equation by dividing all terms by 6:

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

### Step 4: Factor the quadratic equation
Now, we will factor the quadratic equation:

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

### Step 5: Find the critical points
Setting each factor to zero gives us the critical points:

\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Step 6: Determine the intervals
The critical points divide the number line into intervals. We will test the intervals \( (-\infty, -2) \), \( (-2, -1) \), and \( (-1, \infty) \).

### Step 7: Test the intervals
1. **Interval \( (-\infty, -2) \)**: Choose \( x = -3 \)
\[ f'(-3) = 6(-3)^2 + 18(-3) + 12 = 54 - 54 + 12 = 12 \quad (\text{positive}) \]

2. **Interval \( (-2, -1) \)**: Choose \( x = -1.5 \)
\[ f'(-1.5) = 6(-1.5)^2 + 18(-1.5) + 12 = 13.5 - 27 + 12 = -1.5 \quad (\text{negative}) \]

3. **Interval \( (-1, \infty) \)**: Choose \( x = 0 \)
\[ f'(0) = 6(0)^2 + 18(0) + 12 = 12 \quad (\text{positive}) \]

### Step 8: Conclusion
From our tests, we conclude:
- The function is **increasing** on the intervals \( (-\infty, -2) \) and \( (-1, \infty) \).
- The function is **decreasing** on the interval \( (-2, -1) \).

### Final Answer
(a) The function is increasing on the intervals \( (-\infty, -2) \) and \( (-1, \infty) \).

(b) The function is decreasing on the interval \( (-2, -1) \).

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