Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=2x^(3)-3x^(2)-36x+7`.

Answer

To determine the intervals in which the function \( f(x) = 2x^3 - 3x^2 - 36x + 7 \) is strictly increasing or strictly decreasing, we will follow these steps:

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

\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 36x + 7) \]

Using the power rule, we differentiate each term:

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

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

\[ 6x^2 - 6x - 36 = 0 \]

### Step 3: Simplify the equation
We can simplify this equation by dividing all terms by 6:

\[ x^2 - x - 6 = 0 \]

### Step 4: Factor the quadratic equation
Now, we will factor the quadratic equation:

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

### Step 5: Solve for \( x \)
Setting each factor to zero gives us the critical points:

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

### Step 6: Determine the sign of the derivative
We will now test the intervals determined by the critical points \( x = -2 \) and \( x = 3 \). The intervals to test are \( (-\infty, -2) \), \( (-2, 3) \), and \( (3, \infty) \).

1. **Interval \( (-\infty, -2) \)**: Choose \( x = -3 \)
\[ f'(-3) = 6(-3)^2 - 6(-3) - 36 = 54 + 18 - 36 = 36 > 0 \quad \text{(increasing)} \]

2. **Interval \( (-2, 3) \)**: Choose \( x = 0 \)
\[ f'(0) = 6(0)^2 - 6(0) - 36 = -36 < 0 \quad \text{(decreasing)} \]

3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \)
\[ f'(4) = 6(4)^2 - 6(4) - 36 = 96 - 24 - 36 = 36 > 0 \quad \text{(increasing)} \]

### Step 7: Summarize the results
From the tests, we find:
- The function is **strictly increasing** on the intervals \( (-\infty, -2) \) and \( (3, \infty) \).
- The function is **strictly decreasing** on the interval \( (-2, 3) \).

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