Derivative Graphs Analysis
Analyze and interpret the graphs of functions and their derivatives, identifying key features such as increasing/decreasing intervals, local extrema, concavity, and points of inflection.
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Derivative Graphs Analysis
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Carefully analyze the provided graphs and answer the questions that follow. Show all your work for short answer questions.
The graph below shows the derivative f'(x) of a function f(x).
1. On what intervals is f(x) increasing?
2. At what x-values does f(x) have local extrema? Classify each as a local maximum or minimum.
3. If the graph of f''(x) is given below, which statement best describes the concavity of f(x) at x = 1?
f(x) is concave up
f(x) is concave down
f(x) has a point of inflection
f(x) is increasing
4. A point of inflection occurs where the concavity of a function changes. This happens when f''(x) changes sign, or when f''(x) = 0 and f''(x) .
5. If f'(x) has a local maximum or minimum, then f(x) has a at that x-value.
6. If f'(x) > 0 on an interval, then f(x) is concave up on that interval.
True
False
7. A function can have a local maximum where f'(x) is undefined.
True
False
8. Given the graph of f'(x) below, sketch a possible graph of f(x) on the coordinate plane provided. Assume f(0) = 0.