First Derivative Test Worksheet
Practice applying the first derivative test to find increasing/decreasing intervals and local extrema of functions.
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First Derivative Test Worksheet
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Read each question carefully and show all your work. Use the first derivative test to determine the intervals where the function is increasing or decreasing, and to find any local maximum or minimum values.
1. For the function f(x) = x³ - 6x² + 5, find the intervals where f(x) is increasing and decreasing.
2. Which of the following is a local maximum for the function g(x) = -x³ + 3x² - 2?
(0, -2)
(2, 2)
(1, 0)
(3, -2)
3. The critical points of h(x) = 2x³ - 3x² - 12x + 1 are x = and x = . At these points, h(x) has a local and a local , respectively.
4. Consider the function k(x) = x⁴ - 8x² + 1. Use the first derivative test to determine the intervals of increasing/decreasing and locate all local extrema. Sketch a number line to illustrate your findings.
5. If f'(c) = 0, then f(x) must have a local maximum or minimum at x = c.
True
False
6. A function is increasing on an interval if its first derivative is positive on that interval.
True
False
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