The interval in which the function `f(x) =2x^(3) + 9x^(2) + 12x + 20` is strictly increasing, is:

Options

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

Answer

To determine the interval in which the function \( f(x) = 2x^3 + 9x^2 + 12x + 20 \) is strictly increasing, we need to follow these steps:

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

\[ f'(x) = \frac{d}{dx}(2x^3 + 9x^2 + 12x + 20) \]

Using the power rule, we differentiate each term:

- The derivative of \( 2x^3 \) is \( 6x^2 \).
- The derivative of \( 9x^2 \) is \( 18x \).
- The derivative of \( 12x \) is \( 12 \).
- The derivative of a constant \( 20 \) is \( 0 \).

Thus, we have:

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

### Step 2: Set the derivative greater than zero
To find where the function is increasing, we set the derivative greater than zero:

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

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

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

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

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

So, we rewrite the inequality:

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

### Step 5: Determine the critical points
The critical points from the factors are:

- \( x + 1 = 0 \) gives \( x = -1 \)
- \( x + 2 = 0 \) gives \( x = -2 \)

### Step 6: Test intervals around the critical points
We will test the intervals determined by the critical points \( -2 \) and \( -1 \):

1. **Interval \( (-\infty, -2) \)**: Choose \( x = -3 \)
\[ (-3 + 1)(-3 + 2) = (-2)(-1) = 2 > 0 \quad \text{(True)} \]

2. **Interval \( (-2, -1) \)**: Choose \( x = -1.5 \)
\[ (-1.5 + 1)(-1.5 + 2) = (-0.5)(0.5) = -0.25 < 0 \quad \text{(False)} \]

3. **Interval \( (-1, \infty) \)**: Choose \( x = 0 \)
\[ (0 + 1)(0 + 2) = (1)(2) = 2 > 0 \quad \text{(True)} \]

### Step 7: Write the solution
The function \( f(x) \) is strictly increasing in the intervals where the inequality holds true:

\[ (-\infty, -2) \cup (-1, \infty) \]

### Final Answer
The interval in which the function \( f(x) \) is strictly increasing is:

\[ (-\infty, -2) \cup (-1, \infty) \]

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