Graph the following function \( f(x) \). Estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima. \( f(x)=x^{2}+2 x-8 \) A. The function \( f \) is increasing on the interval(s) (Type your answer in interval notation. Use a comma to separate answers as needed.) B. There is no interval on which the function \( f(x) \) is increasing. Determine on which interval(s) \( f(x) \) is decreasing. Select the correct choice

To analyze the function \( f(x) = x^2 + 2x - 8 \), we can start by finding its critical points and determining where it is increasing or decreasing. ### Step 1: Find the derivative The first step is to find the derivative of the function: \[ f'(x) = \frac{d}{dx}(x^2 + 2x - 8) = 2x + 2 \] ### Step 2: Set the derivative to zero Next, we set the derivative equal to zero to find the critical points: \[ 2x + 2 = 0 \] \[ 2x = -2 \] \[ x = -1 \] ### Step 3: Determine the sign of the derivative Now we will test the intervals around the critical point \( x = -1 \) to determine where the function is increasing or decreasing. - **Interval 1**: \( (-\infty, -1) \) - Choose a test point, e.g., \( x = -2 \): \[ f'(-2) = 2(-2) + 2 = -4 + 2 = -2 \quad (\text{negative}) \] Thus, \( f(x) \) is decreasing on \( (-\infty, -1) \). - **Interval 2**: \( (-1, \infty) \) - Choose a test point, e.g., \( x = 0 \): \[ f'(0) = 2(0) + 2 = 0 + 2 = 2 \quad (\text{positive}) \] Thus, \( f(x) \) is increasing on \( (-1, \infty) \). ### Step 4: Identify relative maxima and minima Since the function changes from decreasing to increasing at \( x = -1 \), this point is a relative minimum. ### Step 5: Calculate the value of the function at the critical point To find the value of the function at the critical point: \[ f(-1) = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9 \] So, the relative minimum is at the point \( (-1, -9) \). ### Summary of Results A. The function \( f \) is increasing on the interval(s): \[ (-1, \infty) \] B. The function \( f(x) \) is decreasing on the interval(s): \[ (-\infty, -1) \] ### Final Answers A. \( (-1, \infty) \) B. \( (-\infty, -1) \)