Question

The figure to the right shows the graph of f', the derivative of the function f. The graph of f'(x) has horizontal tangent lines at x=-1, x=1 and x=3. Your justification must use the graph given to you.

a. For what value(s) of x does f have a relative minimum? Justify your answer.
b. For what value(s) of x does f have a relative maximum? Justify your answer.
c. On what interval(s) is the graph of f concave upward? Justify your answer.
d. Use the information found in parts (a), (b) and (c) and the fact that f(-3)=0 to sketch a possible graph of f on the axes below.

Answer 1

Explanation:

In AI text-based model, I am unable to view or analyze images or graphs. Therefore, I cannot provide a direct analysis of the given graph or answer the questions based on it.

However, I can provide some general information regarding relative minimums, relative maximums, concavity, and sketching a possible graph based on given conditions.

a. For a function f to have a relative minimum at a specific point, the derivative f'(x) must change from negative to positive at that point. This means that the graph of f' has a decreasing slope approaching that point and an increasing slope afterward. You can identify potential relative minimum points on the graph of f' by looking for places where the graph changes from a decreasing slope to an increasing slope.

b. Similarly, for a function f to have a relative maximum at a specific point, the derivative f'(x) must change from positive to negative at that point. This means that the graph of f' has an increasing slope approaching that point and a decreasing slope afterward. You can identify potential relative maximum points on the graph of f' by looking for places where the graph changes from an increasing slope to a decreasing slope.

c. The graph of f is concave upward when the second derivative f''(x) is positive. To determine the concavity of f, you need to examine the graph of f''(x) or look for intervals where the slope of f' is increasing.

d. To sketch a possible graph of f based on given conditions and the fact that f(-3) = 0, you can start by considering the information obtained from parts (a), (b), and (c). Use the relative minimum and maximum points identified to plot the corresponding points on the graph. Additionally, consider the concavity information to determine how the graph curves in different regions. Finally, use the fact that f(-3) = 0 to include the point (-3, 0) on the graph.

Please note that without the actual graph or specific equations, the sketching of a possible graph is speculative and may vary depending on the specific characteristics of the function f.