Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=2x^(3)-15x^(2)+36x+17.`

Answer

To determine the intervals in which the function \( f(x) = 2x^3 - 15x^2 + 36x + 17 \) is strictly increasing or strictly decreasing, we will follow these steps:

### Step 1: Find the derivative \( f'(x) \)
To find the intervals of increase and decrease, we first need to compute the derivative of the function.

\[ f'(x) = \frac{d}{dx}(2x^3 - 15x^2 + 36x + 17) \]

Using the power rule, we differentiate each term:

- The derivative of \( 2x^3 \) is \( 6x^2 \).
- The derivative of \( -15x^2 \) is \( -30x \).
- The derivative of \( 36x \) is \( 36 \).
- The derivative of the constant \( 17 \) is \( 0 \).

Thus, we have:

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

### Step 2: Factor the derivative
Next, we will factor \( f'(x) \) to find the critical points:

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

Now, we need to factor the quadratic \( x^2 - 5x + 6 \):

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

So, we can write:

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

### Step 3: Find critical points
To find the critical points, we set \( f'(x) = 0 \):

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

This gives us:

\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Step 4: Test intervals around critical points
The critical points divide the number line into intervals. We will test the sign of \( f'(x) \) in the intervals \( (-\infty, 2) \), \( (2, 3) \), and \( (3, \infty) \).

1. **Interval \( (-\infty, 2) \)**: Choose \( x = 0 \)
\[ f'(0) = 6(0 - 2)(0 - 3) = 6(-2)(-3) = 36 > 0 \]
So, \( f(x) \) is **increasing** on \( (-\infty, 2) \).

2. **Interval \( (2, 3) \)**: Choose \( x = 2.5 \)
\[ f'(2.5) = 6(2.5 - 2)(2.5 - 3) = 6(0.5)(-0.5) = -1.5 < 0 \]
So, \( f(x) \) is **decreasing** on \( (2, 3) \).

3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \)
\[ f'(4) = 6(4 - 2)(4 - 3) = 6(2)(1) = 12 > 0 \]
So, \( f(x) \) is **increasing** on \( (3, \infty) \).

### Step 5: Conclusion
From our analysis, we can conclude:

- The function \( f(x) \) is **strictly increasing** on the intervals \( (-\infty, 2) \) and \( (3, \infty) \).
- The function \( f(x) \) is **strictly decreasing** on the interval \( (2, 3) \).

### Summary of Intervals
- **Strictly Increasing**: \( (-\infty, 2) \) and \( (3, \infty) \)
- **Strictly Decreasing**: \( (2, 3) \)