Find the intervals in which the function `f(x)=2x^(3)-9x^(2)+12x+29` is :
(i) monotonic increasing (ii) monotonic decreasing.

Answer

To find the intervals in which the function \( f(x) = 2x^3 - 9x^2 + 12x + 29 \) is monotonic increasing or decreasing, we will follow these steps:

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

\[ f'(x) = \frac{d}{dx}(2x^3 - 9x^2 + 12x + 29) \]

Using the power rule of differentiation, we get:

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

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

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

Dividing the entire equation by 6 simplifies it:

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

### Step 3: Factor the quadratic equation
Now we will factor the quadratic equation:

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

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 4: Determine the sign of the derivative in the intervals
We will analyze the sign of \( f'(x) \) in the intervals determined by the critical points \( x = 1 \) and \( x = 2 \). The intervals to test are \( (-\infty, 1) \), \( (1, 2) \), and \( (2, \infty) \).

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

- **Interval \( (1, 2) \)**: Choose \( x = 1.5 \)
\[ f'(1.5) = 6(1.5)^2 - 18(1.5) + 12 = 6(2.25) - 27 + 12 = 13.5 - 27 + 12 = -1.5 \quad (\text{negative}) \]

- **Interval \( (2, \infty) \)**: Choose \( x = 3 \)
\[ f'(3) = 6(3)^2 - 18(3) + 12 = 54 - 54 + 12 = 12 \quad (\text{positive}) \]

### Step 5: Summarize the intervals
From our analysis:
- \( f'(x) > 0 \) in the intervals \( (-\infty, 1) \) and \( (2, \infty) \) (monotonic increasing).
- \( f'(x) < 0 \) in the interval \( (1, 2) \) (monotonic decreasing).

### Final Answer
(i) The function \( f(x) \) is monotonic increasing in the intervals \( (-\infty, 1) \) and \( (2, \infty) \).

(ii) The function \( f(x) \) is monotonic decreasing in the interval \( (1, 2) \).

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