Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=x^(3)-6x^(2)+9x+8`

Answer

To determine the intervals in which the function \( f(x) = x^3 - 6x^2 + 9x + 8 \) is strictly increasing or strictly decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
The first step is to differentiate the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(6x^2) + \frac{d}{dx}(9x) + \frac{d}{dx}(8) \]

Calculating the derivatives:

- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( 6x^2 \) is \( 12x \).
- The derivative of \( 9x \) is \( 9 \).
- The derivative of the constant \( 8 \) is \( 0 \).

Thus, we have:

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

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

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

Dividing the entire equation by 3:

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

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

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

Setting each factor to zero gives us the critical points:

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Step 4: Test intervals around the critical points
We will now test the intervals defined by the critical points \( x = 1 \) and \( x = 3 \) to determine where the function is increasing or decreasing. The intervals are:

- \( (-\infty, 1) \)
- \( (1, 3) \)
- \( (3, \infty) \)

We will choose test points from each interval:

1. **For the interval \( (-\infty, 1) \)**, choose \( x = 0 \):
\[ f'(0) = 3(0)^2 - 12(0) + 9 = 9 > 0 \quad \text{(increasing)} \]

2. **For the interval \( (1, 3) \)**, choose \( x = 2 \):
\[ f'(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3 < 0 \quad \text{(decreasing)} \]

3. **For the interval \( (3, \infty) \)**, choose \( x = 4 \):
\[ f'(4) = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9 > 0 \quad \text{(increasing)} \]

### Step 5: Conclusion
Based on our tests, we can conclude:

- The function \( f(x) \) is **strictly increasing** on the intervals \( (-\infty, 1) \) and \( (3, \infty) \).
- The function \( f(x) \) is **strictly decreasing** on the interval \( (1, 3) \).

### Final Answer
- **Strictly Increasing**: \( x \in (-\infty, 1) \cup (3, \infty) \)
- **Strictly Decreasing**: \( x \in (1, 3) \)