Draw a graph to match the description given. \( f(x) \) has a positive derivative over \( (-\infty,-1) \) and \( (-1,3) \), a negative derivative over \( (3, \infty) \), and a derivative equal to 0 at \( x=-1 \).

To create a graph of the function \( f(x) \) based on the given description, we need to consider the behavior of the function and its derivative in the specified intervals. 1. **Positive Derivative**: - In the intervals \( (-\infty, -1) \) and \( (-1, 3) \), the function \( f(x) \) is increasing because the derivative \( f'(x) > 0 \). 2. **Derivative Equal to Zero**: - At \( x = -1 \), the derivative \( f'(-1) = 0 \). This indicates that \( x = -1 \) is a critical point, which could be a local minimum or maximum. 3. **Negative Derivative**: - In the interval \( (3, \infty) \), the function \( f(x) \) is decreasing because the derivative \( f'(x) < 0 \). ### Sketching the Graph - Start by plotting the point at \( x = -1 \). Since the function is increasing before and after this point, we can assume that \( f(-1) \) is a local maximum. - As \( x \) approaches \( -1 \) from the left, the function should be increasing, so it will rise towards \( f(-1) \). - After \( x = -1 \), the function continues to increase until it reaches \( x = 3 \). At this point, we can assume that the function is still increasing but at a decreasing rate, possibly reaching a local maximum at \( x = 3 \). - After \( x = 3 \), the function starts to decrease, indicating that it will slope downwards as \( x \) moves towards \( \infty \). ### Summary of the Graph - The graph will start from the left, increasing towards the point \( (-1, f(-1)) \). - It will continue to increase until it reaches a point at \( (3, f(3)) \) where it may reach a local maximum. - After \( x = 3 \), the graph will start to decrease as \( x \) moves towards \( \infty \). ### Example of a Possible Function A simple example of such a function could be: \[ f(x) = -\frac{1}{2}(x + 1)^2 + 4 \quad \text{for } x \leq 3 \] This function has a maximum at \( x = -1 \) and increases until \( x = 3 \), then it can be modified to decrease after \( x = 3 \). ### Final Note While I cannot draw the graph directly here, you can visualize it based on the description above or use graphing software to plot the function based on the characteristics outlined.