Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=2x^(3)-12x^(2)+18x+5`

Answer

To determine the intervals in which the function \( f(x) = 2x^3 - 12x^2 + 18x + 5 \) is strictly increasing or strictly decreasing, we will follow these steps:

### Step 1: Find the first derivative of the function
The first derivative \( f'(x) \) will help us determine the intervals of increase and decrease.

\[ f'(x) = \frac{d}{dx}(2x^3 - 12x^2 + 18x + 5) \]
Using the power rule, we differentiate each term:

\[ f'(x) = 6x^2 - 24x + 18 \]

### Step 2: Set the first derivative greater than zero for increasing intervals
To find where the function is increasing, we set the first derivative greater than zero:

\[ 6x^2 - 24x + 18 > 0 \]

### Step 3: Simplify the inequality
We can factor out the common term 6:

\[ 6(x^2 - 4x + 3) > 0 \]

Dividing both sides by 6 (since 6 is positive, the inequality remains the same):

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

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

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

So, we have:

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

### Step 5: Determine the critical points
The critical points from the factors are \( x = 1 \) and \( x = 3 \). These points divide the number line into intervals.

### Step 6: Test the intervals
We will test the sign of \( (x - 1)(x - 3) \) in the intervals:
1. \( (-\infty, 1) \)
2. \( (1, 3) \)
3. \( (3, \infty) \)

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

- **Interval \( (1, 3) \)**: Choose \( x = 2 \)
\[ (2 - 1)(2 - 3) = (1)(-1) = -1 < 0 \quad \text{(Decreasing)} \]

- **Interval \( (3, \infty) \)**: Choose \( x = 4 \)
\[ (4 - 1)(4 - 3) = (3)(1) = 3 > 0 \quad \text{(Increasing)} \]

### Step 7: Summarize the intervals
From our tests, we 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 Result:
- **Increasing Intervals**: \( (-\infty, 1) \cup (3, \infty) \)
- **Decreasing Interval**: \( (1, 3) \)