Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=2x^(3)-15x^(2)+36x+6`

Answer

To determine the intervals in which the function \( f(x) = 2x^3 - 15x^2 + 36x + 6 \) 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 \( f(x) \).

\[ f'(x) = \frac{d}{dx}(2x^3 - 15x^2 + 36x + 6) \]

Using the power rule for differentiation:

\[ f'(x) = 6x^2 - 30x + 36 \]

### Step 2: Factor the derivative
Next, we will factor the derivative to find the critical points.

\[ f'(x) = 6(x^2 - 5x + 6) \]

Now we can factor the quadratic expression:

\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

Thus, we have:

\[ f'(x) = 6(x - 2)(x - 3) \]

### Step 3: Find the critical points
To find the critical points, we set the derivative equal to zero:

\[ 6(x - 2)(x - 3) = 0 \]

This gives us the critical points:

\[ x = 2 \quad \text{and} \quad x = 3 \]

### Step 4: Determine the sign of the derivative
We will analyze the sign of \( f'(x) \) in the intervals determined by the critical points \( x = 2 \) and \( x = 3 \). The intervals to test are:

1. \( (-\infty, 2) \)
2. \( (2, 3) \)
3. \( (3, \infty) \)

**Test the interval \( (-\infty, 2) \)**:
Choose \( x = 0 \):

\[ f'(0) = 6(0 - 2)(0 - 3) = 6(-2)(-3) = 36 > 0 \]

**Test the interval \( (2, 3) \)**:
Choose \( x = 2.5 \):

\[ f'(2.5) = 6(2.5 - 2)(2.5 - 3) = 6(0.5)(-0.5) = -1.5 < 0 \]

**Test the interval \( (3, \infty) \)**:
Choose \( x = 4 \):

\[ f'(4) = 6(4 - 2)(4 - 3) = 6(2)(1) = 12 > 0 \]

### Step 5: Conclusion on intervals
From our tests, we conclude:

- \( f'(x) > 0 \) in the intervals \( (-\infty, 2) \) and \( (3, \infty) \), meaning \( f(x) \) is strictly increasing in these intervals.
- \( f'(x) < 0 \) in the interval \( (2, 3) \), meaning \( f(x) \) is strictly decreasing in this interval.

### Final Answer
- **Strictly Increasing**: \( (-\infty, 2) \) and \( (3, \infty) \)
- **Strictly Decreasing**: \( (2, 3) \)