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

Answer

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

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first need to find its derivative \( f'(x) \).

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

Using the power rule for differentiation:

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

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

\[ f'(x) = 3x(x + 2) \]

### Step 3: Set the derivative to zero
To find the critical points where the function changes from increasing to decreasing or vice versa, we set the derivative equal to zero:

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

This gives us two critical points:

\[ 3x = 0 \implies x = 0 \]
\[ x + 2 = 0 \implies x = -2 \]

### Step 4: Determine intervals for testing
The critical points divide the number line into intervals. We will test the sign of \( f'(x) \) in the intervals:

1. \( (-\infty, -2) \)
2. \( (-2, 0) \)
3. \( (0, \infty) \)

### Step 5: Test the intervals
We will choose test points from each interval to determine whether \( f'(x) \) is positive or negative.

- **Interval \( (-\infty, -2) \)**: Choose \( x = -3 \)
\[ f'(-3) = 3(-3)((-3) + 2) = 3(-3)(-1) = 9 \quad (\text{positive}) \]

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

- **Interval \( (0, \infty) \)**: Choose \( x = 1 \)
\[ f'(1) = 3(1)((1) + 2) = 3(1)(3) = 9 \quad (\text{positive}) \]

### Step 6: Conclusion about increasing and decreasing intervals
From our tests, we can conclude:
- \( f'(x) > 0 \) in the intervals \( (-\infty, -2) \) and \( (0, \infty) \), so \( f(x) \) is **strictly increasing** in these intervals.
- \( f'(x) < 0 \) in the interval \( (-2, 0) \), so \( f(x) \) is **strictly decreasing** in this interval.

### Final Answer
- **Strictly Increasing**: \( (-\infty, -2) \) and \( (0, \infty) \)
- **Strictly Decreasing**: \( (-2, 0) \)

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