4.2 Critical Points Local Maxima and Local Minima

Thefirstderivativetest

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Unformatted text preview: How do we know if there is a local maximum, local minimum or neither at a critical point? Answer: Combining this with the properties of increasing and decreasing functions, we have a first derivative test for local extrema. The First Derivative Test Suppose c is a critical number of a function f(x). If f'(x) changes sign from negative to positive at x = c, then f(x) has a local minimum at c. If f'(x) changes sign from positive to negative at x = c, then f(x) has a local maximum c. Note: * If f'(x) doe not change sign at c, then f(x) has neither a local maximum nor a local minimum at c. 1 4.2 Critical Points Local Maxima and Local Minima.notebook May 15, 2009 Example 1 (a) Determine the critical numbers of the function f(x) = ‐2x3 + 9x2 + 4. (b) What are the critical points? (c) Determine whether each critical point is a local maximum, local minimum, or neither. (d) Sketch a graph of the function. Example 2 Determine the local extrema of the function f(x) = x3, at (c, f(c)), where f'(c) = 0. 2 4.2 Critical Points Local Maxima and Local Minima.notebook May 15, 2009 Example 3 For the function f(x) = I x + 2 I, determine the critical numbers. Example 4 Given the graph of a polynomial function y = f(x), graph y = f'(x). y 0 x 3 4.2 Critical Points Local Maxima and Local Minima.notebook May 15, 2009 Work Assigned: page 178 # 3, 4, 5c, 7e, 9, 12 ‐ 14, 17 4...
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