If `f(x) = x^(3) - 6x^(2) + 9x + 3` be a decreasing function, then x lies in

Options

A. `(-oo, -1) cap (3, oo)`
B. (1, 3)
C. `(3, oo)`
D. None of these

Answer

To determine the interval in which the function \( f(x) = x^3 - 6x^2 + 9x + 3 \) is decreasing, we need to follow these steps:

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

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

Using the power rule for differentiation, we get:

\[ f'(x) = 3x^2 - 12x + 9 \]

### Step 2: Set the derivative less than zero
For the function to be decreasing, we need:

\[ f'(x) < 0 \]

This translates to:

\[ 3x^2 - 12x + 9 < 0 \]

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

\[ x^2 - 4x + 3 < 0 \]

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

\[ x^2 - 4x + 3 = (x - 1)(x - 3) \]

Thus, the inequality becomes:

\[ (x - 1)(x - 3) < 0 \]

### Step 5: Determine the intervals
To find the intervals where this product is negative, we identify the critical points, which are \( x = 1 \) and \( x = 3 \). We can test the intervals determined by these points:

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

2. **Interval \( (1, 3) \)**: Choose \( x = 2 \)
\[ (2 - 1)(2 - 3) = -1 \quad \text{(negative)} \]

3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \)
\[ (4 - 1)(4 - 3) = 3 \quad \text{(positive)} \]

### Step 6: Conclusion
The function \( f(x) \) is decreasing in the interval \( (1, 3) \).

Thus, the final answer is:
\[ x \text{ lies in } (1, 3) \]