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 has a relative maximum and the maximum value is \( \square \) at \( x=\square \). B. The function has no relative maximum.

To analyze the function \( f(x) = x^2 + 2x - 8 \), we can start by finding its critical points, which will help us determine the intervals of increase and decrease, as well as any relative maxima or minima. ### Step 1: Find the derivative First, we find the derivative of the function: \[ f'(x) = 2x + 2 \] ### 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 nature of the critical point To determine whether this critical point is a maximum or minimum, we can use the second derivative test. We find the second derivative: \[ f''(x) = 2 \] Since \( f''(x) = 2 > 0 \), the function is concave up at all points, indicating that the critical point at \( x = -1 \) is a relative minimum. ### Step 4: Find the value of the function at the critical point Now we can find the value of the function at this critical point: \[ f(-1) = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9 \] ### Step 5: Determine intervals of increase and decrease To find the intervals where the function is increasing or decreasing, we can analyze the sign of the first derivative \( f'(x) \): - For \( x < -1 \): \( f'(-2) = 2(-2) + 2 = -2 \) (decreasing) - For \( x > -1 \): \( f'(0) = 2(0) + 2 = 2 \) (increasing) Thus, the function is: - Decreasing on the interval \( (-\infty, -1) \) - Increasing on the interval \( (-1, \infty) \) ### Summary of findings - The function has a relative minimum at \( x = -1 \) with a minimum value of \( -9 \). - There are no relative maxima. ### Final Answers A. The function has a relative maximum and the maximum value is \( \square \) at \( x=\square \). B. The function has no relative maximum. Thus, the correct answer is: - A. \( \square \) (no relative maximum) - B. The function has no relative maximum.