Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=3x^(4)-4x^(3)-12x^(2)+5.`

Answer

To determine the intervals in which the function \( f(x) = 3x^4 - 4x^3 - 12x^2 + 5 \) is strictly increasing or strictly decreasing, we will follow these steps:

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

\[ f'(x) = \frac{d}{dx}(3x^4 - 4x^3 - 12x^2 + 5) \]

Using the power rule of differentiation:

\[ f'(x) = 12x^3 - 12x^2 - 24x \]

### Step 2: Factor the derivative
Next, we can factor out the common term from the derivative:

\[ f'(x) = 12x(x^2 - x - 2) \]

Now, we can factor the quadratic expression \( x^2 - x - 2 \):

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

Thus, the derivative can be rewritten as:

\[ f'(x) = 12x(x - 2)(x + 1) \]

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

\[ 12x(x - 2)(x + 1) = 0 \]

This gives us the critical points:

\[ x = 0, \quad x = 2, \quad x = -1 \]

### Step 4: Determine the sign of the derivative
We will analyze the sign of \( f'(x) \) in the intervals determined by the critical points: \( (-\infty, -1) \), \( (-1, 0) \), \( (0, 2) \), and \( (2, \infty) \).

1. **Interval \( (-\infty, -1) \)**:
- Choose \( x = -2 \):
\[ f'(-2) = 12(-2)(-2 - 2)(-2 + 1) = 12(-2)(-4)(-1) = -96 \quad (\text{Negative}) \]

2. **Interval \( (-1, 0) \)**:
- Choose \( x = -0.5 \):
\[ f'(-0.5) = 12(-0.5)(-0.5 - 2)(-0.5 + 1) = 12(-0.5)(-2.5)(0.5) = 7.5 \quad (\text{Positive}) \]

3. **Interval \( (0, 2) \)**:
- Choose \( x = 1 \):
\[ f'(1) = 12(1)(1 - 2)(1 + 1) = 12(1)(-1)(2) = -24 \quad (\text{Negative}) \]

4. **Interval \( (2, \infty) \)**:
- Choose \( x = 3 \):
\[ f'(3) = 12(3)(3 - 2)(3 + 1) = 12(3)(1)(4) = 144 \quad (\text{Positive}) \]

### Step 5: Summarize the intervals
From our analysis, we can summarize the behavior of the function:

- **Increasing**: \( (-1, 0) \) and \( (2, \infty) \)
- **Decreasing**: \( (-\infty, -1) \) and \( (0, 2) \)

### Final Answer
The function \( f(x) \) is:
- **Strictly increasing** on the intervals \( (-1, 0) \) and \( (2, \infty) \).
- **Strictly decreasing** on the intervals \( (-\infty, -1) \) and \( (0, 2) \).