To sketch a graph of the function f(x) given the information provided, we need to analyze each piece of information and understand its implications on the graph. Let’s start by examining the domain of f.

The domain of f is given as (-∞, -1) ∪ (-1, ∞), which means that the function is defined for all real numbers except for x = -1. This indicates that there might be a vertical asymptote at x = -1, as the function approaches infinity as x approaches -1.

Next, we can look at the given function values f(0) and f(1). We know that f(0) = 0 and f(1) = 1/2. This information allows us to plot two points on the graph, (0, 0) and (1, 1/2).

Moving on to the derivatives of f, we are given that f′(-1) is undefined, f′(0) = 0, and f′(x) has specific properties on different intervals. Let’s analyze these derivatives to determine the increasing and decreasing intervals of f.

First, f′(-1) is undefined, indicating that there might be a vertical tangent or a sharp corner at x = -1. This information suggests that there might be a local extremum at this point. We will keep this in mind when sketching the graph.

Second, f′(0) = 0, which means that at x = 0, the function has a horizontal tangent. This suggests that there might be a local extremum or a point of inflection at x = 0.

Based on the given information about f′(x), we can conclude the following about the increasing and decreasing intervals:

– f′(x) > 0 on (-∞, -1) and (0, ∞): This indicates that the function is increasing on these intervals.

– f′(x) < 0 on (-1, 0): This indicates that the function is decreasing on this interval.
Next, let's analyze the second derivatives f′′(x) and determine the concavity of the function.
We are given that f′′(-1) is undefined and f′′(1) = 0. We can conclude the following about the concavity of f:
- f′′(x) > 0 on (-∞, -1) and (-1, 1): This indicates that the function is concave up on these intervals.

– f′′(x) < 0 on (1, ∞): This indicates that the function is concave down on this interval.
Now that we have analyzed all the given information, we can start sketching the graph of f(x).
First, we plot the points (0, 0) and (1, 1/2) as mentioned earlier. These two points help us understand the behavior of the graph near x = 0 and x = 1.
Next, we know that there might be a vertical asymptote at x = -1. Since the function approaches infinity as x approaches -1, we can draw a vertical dashed line to represent the asymptote.
Based on the increasing and decreasing intervals of f(x), we can determine the direction of the graph. The function is increasing on (-∞, -1) and (0, ∞), so the graph should be rising in those intervals. On the interval (-1, 0), the function is decreasing, so the graph should be descending in that interval.
Similarly, based on the concavity of the function, we can determine the shape of the graph. The function is concave up on (-∞, -1) and (-1, 1), so the graph should be seeing an upward curvature in those intervals. On the interval (1, ∞), the function is concave down, so the graph should be seeing a downward curvature in that interval.
Continued...