Increasing, Decreasing, Constant Objectives: Be able to apply the first derivative test to find relative extrema of a function Critical Vocabulary: Increasing, Decreasing, Constant Warm Up: Determine (using calculus) where each function is increasing, decreasing, and constant. 1. f(x) = 2x3 - 3x2 - 36x + 14 2. f(x) = (x2 - 4)2/3 3.

This lesson is going to discuss the behavior of a critical number. The previous lesson discussed the behavior of an interval (increasing, decreasing, and constant). This lesson is going to discuss the behavior of a critical number. If an interval changes from increasing to decreasing (or decreasing to increasing), then what is happening at the critical number that separates the intervals? Interval (-∞, 0) (0, 1) (1, ∞) Test Value x = -5 x = ½ x = 25 f’(x) = (+) f’(x) = (-) f’(x) = (+) Sign of f’(x) Conclusion Increasing Decreasing Increasing 1 Relative Max Relative Min

1. Use increasing/decreasing test to find the intervals. Let c be a critical number of a function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows: If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f. If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f. If f’(x) does not change signs at c, then f(c) is neither a relative maximum nor relative minimum of f. Summary: 1. Use increasing/decreasing test to find the intervals. Use the First Derivative Test to determine if a critical number is a relative max or min.

Relative Min Relative Max Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results. Example 1: f(x) = 2x3 - 3x2 - 36x + 14 f’(x) = 6x2 - 6x - 36 0 = 6x2 - 6x - 36 0 = 6(x - 3)(x + 2) x = 3 and x = -2 Interval (-∞, -2) (-2, 3) (3, ∞) Test Value x = -4 x = 0 x = 4 Increasing: (-∞, -2) (3, ∞) Decreasing: (-2, 3) Sign f’(x) f’(x) = (+) f’(x) = (-) f’(x) = (+) Constant: Never Conclusion Increasing Decreasing Increasing -2 3 Relative Min Relative Max Relative Max: x = -2 Relative Min: x = 3

Relative Min Relative Min Relative Max Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results. Example 2: f(x) = (x2 - 4)2/3 4x = 0 x = 0 Undefined at x = 2 , x = -2 Interval (-∞, -2) (-2, 0) (0, 2) (2, ∞) Increasing: (-2, 0) (2, ∞) Test Value x = -3 x = -1 x = 1 x = 3 Decreasing: (-∞, -2) (0, 2) Sign f’(x) f’(x) = (-) f’(x) = (+) f’(x) = (-) f’(x) = (+) Constant: Never Conclusion Decreasing Increasing Decreasing Increasing Relative Max: x = 0 -2 2 Relative Min: x = 2 x = -2 Relative Min Relative Min Relative Max

Relative Min Relative Min Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results. Example 3: 2x4 – 2 = 0 x = 1 and x = -1 Discontinuity at x = 0 Interval (-∞, -1) (-1, 0) (0, 1) (1, ∞) Increasing: (-1, 0) (1, ∞) Test Value x = -3 x = -½ x = ½ x = 3 Decreasing: (-∞, -1) (0, 1) Sign f’(x) f’(x) = (-) f’(x) = (+) f’(x) = (-) f’(x) = (+) Constant: Never Conclusion Decreasing Increasing Decreasing Increasing Relative Max: None -1 1 Relative Min: x = 1 x = -1 Relative Min Relative Min

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