Find the interval (s) in which f(x) `=-x ^(2) -2x+ 15` is increasing.

Answer

To find the interval in which the function \( f(x) = -x^2 - 2x + 15 \) is increasing, we will follow these steps:

### Step 1: Differentiate the function
We need to find the derivative of the function \( f(x) \).

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

Using the power rule for differentiation, we get:

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

### Step 2: Set the derivative greater than zero
To find where the function is increasing, we set the derivative greater than zero:

\[ f'(x) > 0 \]

This gives us:

\[ -2x - 2 > 0 \]

### Step 3: Solve the inequality
Now, we will solve the inequality:

\[ -2x > 2 \]

Dividing both sides by -2 (remember to reverse the inequality sign):

\[ x < -1 \]

### Step 4: Determine the interval
The function \( f(x) \) is increasing when \( x < -1 \). Therefore, the interval in which the function is increasing is:

\[ (-\infty, -1) \]

### Final Answer
The function \( f(x) = -x^2 - 2x + 15 \) is increasing on the interval \( (-\infty, -1) \).

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