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3.3: st 1 Derivative Test By: Ted feat. G$ and Jameth Increasing vs. Decreasing A function f is increasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2). A function f is decreasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2). In other words, a function is increasing if, as x moves to the right, its graph moves up and decreasing if the graph moves down. Increasing vs. Decreasing Theorem 3.5: Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). • 1. If f’(x) > 0 for all x in (a, b), then f is increasing on [a, b]. • 2. If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b]. • 3. If f’(x) = 0 for all x in (a, b), then f is constant on [a, b]. Increasing vs. Decreasing In simpler terms, Theorem 3.5 states… 1. A positive derivative implies that the function is increasing. 2. A negative derivative implies that the function is decreasing. 3. A zero derivative implies that the function is constant. Example: f(x) = 3 x • Step 1: Find the derivative of f(x). Answer: f’(x)=3x2 – 3x – 2 1.5x Example (continued) Step 2: Graph the derivative. Step 3: Analyze the graph of the derivative. On the intervals (-∞, 0) and (1, ∞), f’(x) is positive and so f(x) must be increasing on those intervals. On the interval (0, 1), however, f’(x) is negative and so f(x) must be decreasing on that interval. Example (continued) •In this comparison of the top graph f(x), and the bottom graph f’(x), not only does Theorem 3.5 hold true, but, in addition, at the two values where f’(x) intersects with the xaxis, there are a relative maximum and minimum at the corresponding value on f(x). st 1 Derivative Test Let c be a critical number of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows: 1. If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f. 2. If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f. 3. If f’(x) does not change sign at c, then f(c) is neither a relative minimum nor a relative maximum. Example: f(x) = 2 x – 6x Find the critical numbers for the function. Step 1: Take the derivative… f’(x) = 2x – 6. Step 2: Set f’(x) equal to zero and then solve for x to find the critical number(s), if any… 2x – 6 = 0, so x = 3. [Note: The graph of f’(x) crosses the xaxis when f’(x) = 0. So the critical numbers of f(x) correspond to x-intercepts of f’(x).] Example (continued) Step 3: Clearly, there is a critical number at x = 3. To find out if this point is a relative minimum or maximum, look back at the graph of f(x). Step 4: Based on the graph, we can see that the critical number is a relative minimum. Example (continued) Note: If you are not given a graph and/or cannot generate one, you can determine if it’s a relative minimum or maximum by plugging in values for x into the derivative function that are very close to the critical number. In the previous example, the critical number was at x = 3. So plug in 2.9 and 3.1 for x into the derivative equation. f’(2.9) = -.2 f’(3.1) = .2 Since it changed from negative to positive, the critical point is a relative minimum of f(x). Made by… Ted “the goods” Murphy Michael G$ Goepfrich Jameth O’Brien