If `f(x)=3x^(2)+6x-9` then

Options

A. f(x) is increasing in (-1, 3)
B. f(x) is decreasing in `(3, infty)`
C. f(x) is increasing in `(-infty, -1)`
D. f(x) is decreasing in `(-infty, -1)`

Explanation

To determine the intervals where the function \( f(x) = 3x^2 + 6x - 9 \) is increasing or decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first need to find its derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(3x^2 + 6x - 9) \]

Using the power rule, we differentiate each term:

\[ f'(x) = 6x + 6 \]

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

\[ 6x + 6 = 0 \]

Solving for \( x \):

\[ 6x = -6 \\ x = -1 \]

### Step 3: Determine the sign of the derivative
Now we will test the intervals around the critical point \( x = -1 \) to determine where the function is increasing or decreasing. The intervals to test are \( (-\infty, -1) \) and \( (-1, \infty) \).

1. **Choose a test point in the interval \( (-\infty, -1) \)**, for example, \( x = -2 \):

\[ f'(-2) = 6(-2) + 6 = -12 + 6 = -6 \quad (\text{negative}) \]

Therefore, \( f(x) \) is decreasing in the interval \( (-\infty, -1) \).

2. **Choose a test point in the interval \( (-1, \infty) \)**, for example, \( x = 0 \):

\[ f'(0) = 6(0) + 6 = 0 + 6 = 6 \quad (\text{positive}) \]

Therefore, \( f(x) \) is increasing in the interval \( (-1, \infty) \).

### Step 4: Summarize the intervals
From our analysis, we conclude that:
- The function \( f(x) \) is **decreasing** on the interval \( (-\infty, -1) \).
- The function \( f(x) \) is **increasing** on the interval \( (-1, \infty) \).

### Final Answer
The function \( f(x) = 3x^2 + 6x - 9 \) is decreasing on \( (-\infty, -1) \) and increasing on \( (-1, \infty) \).

---