Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=20-9x+6x^(2)-x^(3)`

Answer

To determine the intervals in which the function \( f(x) = 20 - 9x + 6x^2 - x^3 \) is strictly increasing or strictly decreasing, we will follow these steps:

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

\[ f'(x) = \frac{d}{dx}(20 - 9x + 6x^2 - x^3) \]

Calculating the derivative term by term:
- The derivative of \( 20 \) is \( 0 \) (constant).
- The derivative of \( -9x \) is \( -9 \).
- The derivative of \( 6x^2 \) is \( 12x \).
- The derivative of \( -x^3 \) is \( -3x^2 \).

Thus, we have:

\[ f'(x) = 0 - 9 + 12x - 3x^2 = -3x^2 + 12x - 9 \]

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

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

Setting each factor to zero gives us the critical points:

\[ x = 1 \quad \text{and} \quad x = 3 \]

### Step 4: Determine the sign of the derivative in the intervals
We will now test the intervals defined by the critical points \( x = 1 \) and \( x = 3 \). The intervals to test are \( (-\infty, 1) \), \( (1, 3) \), and \( (3, \infty) \).

1. **Interval \( (-\infty, 1) \)**: Choose \( x = 0 \):
\[ f'(0) = -3(0)^2 + 12(0) - 9 = -9 \quad (\text{negative}) \]

2. **Interval \( (1, 3) \)**: Choose \( x = 2 \):
\[ f'(2) = -3(2)^2 + 12(2) - 9 = -3(4) + 24 - 9 = -12 + 24 - 9 = 3 \quad (\text{positive}) \]

3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \):
\[ f'(4) = -3(4)^2 + 12(4) - 9 = -3(16) + 48 - 9 = -48 + 48 - 9 = -9 \quad (\text{negative}) \]

### Step 5: Summarize the intervals
From our tests, we can summarize the behavior of \( f(x) \):
- \( f'(x) < 0 \) in the intervals \( (-\infty, 1) \) and \( (3, \infty) \) (strictly decreasing).
- \( f'(x) > 0 \) in the interval \( (1, 3) \) (strictly increasing).

### Final Result
- **Strictly Increasing**: \( (1, 3) \)
- **Strictly Decreasing**: \( (-\infty, 1) \) and \( (3, \infty) \)