Determine for which values of x, the following functions are increasing or decreasing :
`f(x)=-3x^(2)+12x+8.`

Answer

To determine the intervals where the function \( f(x) = -3x^2 + 12x + 8 \) is increasing or decreasing, we will follow these steps:

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

\[ f'(x) = \frac{d}{dx}(-3x^2 + 12x + 8) \]

Using the power rule, we differentiate each term:

- The derivative of \(-3x^2\) is \(-6x\).
- The derivative of \(12x\) is \(12\).
- The derivative of \(8\) is \(0\).

Thus, we have:

\[ f'(x) = -6x + 12 \]

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

\[ -6x + 12 = 0 \]

Solving for \(x\):

\[ -6x = -12 \implies x = 2 \]

### Step 3: Determine intervals for increasing and decreasing
Now we will analyze the sign of \(f'(x)\) around the critical point \(x = 2\). We will test intervals: \( (-\infty, 2) \) and \( (2, \infty) \).

1. **For the interval \( (-\infty, 2) \)**:
- Choose a test point, for example, \(x = 0\):
\[ f'(0) = -6(0) + 12 = 12 > 0 \]
Thus, \(f(x)\) is increasing on \( (-\infty, 2) \).

2. **For the interval \( (2, \infty) \)**:
- Choose a test point, for example, \(x = 3\):
\[ f'(3) = -6(3) + 12 = -6 < 0 \]
Thus, \(f(x)\) is decreasing on \( (2, \infty) \).

### Step 4: Conclusion
From our analysis, we conclude:
- The function \( f(x) \) is **increasing** on the interval \( (-\infty, 2) \).
- The function \( f(x) \) is **decreasing** on the interval \( (2, \infty) \).

### Summary of Results
- Increasing: \( (-\infty, 2) \)
- Decreasing: \( (2, \infty) \)