3.3 Increasing & Decreasing Functions and The First Derivative Test

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1 3.3 Increasing & Decreasing Functions and The First Derivative Test Definitions of Increasing and Decreasing Functions: A funcon f is increasing on an interval if for any two numbers x 1 and x 2 in the interval, x 1 < x 2 implies f(x 1 ) < f(x 2 ). A funcon f is decreasing on an interval if for any two numbers x 1 and x 2 in the interval, x 1 < x 2 implies f(x 1 ) > f(x 2 ). A function is increasing if, as x moves to the right, its graph moves up, and is decreasing if its graph moves down. The function is decreasing on the interval, is constant on the interval (a, b), and is increasing on the interval 1

2 Using the graph of f(x), sketch a possible f '(x). f(x) x Notice where f(x) has tangent lines with slope = 0. x= This is where f '(x) =, so the graph of f '(x) at these points should cross. Where f(x) decreases = slope; therefore, f '(x) and is the x axis. Where f(x) increases = slope; therefore, f '(x) and is the x axis. 2

3 f(x) x f '(x) x 3

4 Summary! If you have horizontal tangents If f(x) is decreasing If f(x) is increasing Where f(x) "turns" from inc/dec or dec/inc, then f '(x) will be a 4

5 Theorem 3.5 Test for Increasing and Decreasing Functions Let f be a funcon that is connuous on the closed interval [ a, b] and differenable 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]. 5

6 Example 1: Find the open intervals on which increasing or decreasing. Note that f is differenable on the enre real number line. To determine the crical numbers of f, set f '(x) equal to zero. is Because there are no points for which f' does not exist, you can conclude that x = and x = are the only crical numbers. 6

7 So, f is increasing on the intervals and decreasing on the interval (0, 1). and 7

8 Steps for Finding Intervals on which a Funcon is Increasing or Decreasing: Let f be connuous on the interval ( a, b). To find the open intervals on which f is increasing or decreasing, use the following steps: 1. Locate the crical numbers of f in (a, b), and use these numbers to determine test intervals. 2. Determine the sign of f '(x) at one test value in each of the intervals. 3. Use Theorem 3.5 to determine whether f is increasing or decreasing on each interval. These guidelines are also valid if the interval (a, b) is replaced by an interval of the form (, b), (a, ), or (, ). 8

9 A function is strictly monotonic on an interval if it is either increasing on the entire interval or decreasing on the entire interval. For instance, the function f(x) = x 3 is strictly monotonic on the entire real number line because it is increasing on the entire real number line. The function below is not strictly monotonic on the entire real number line because it is constant on the interval [0, 1]. 9

10 The First Derivative Test Once you've found the intervals on which a function is increasing or decreasing, it is not difficult to locate the relative extrema of the function. The function has a relative maximum at the point (0, 0) because f is increasing immediately to the left of x = 0 and decreasing immediately to the right of x = 0. Similarly, the opposite is true for the relative minimum. 10

11 Theorem 3.6 The First Derivave Test Let c be a crical number of a funcon f that is connuous on an open interval I containing c. If f is differenable on the interval, except possibly at c, then f(c) can be classified as follows: 1. If f '(x) changes from negave to posive at c, then f has a relave (local) minimum at (c, f(c)). 2. If f '(x) changes from posive to negave at c, then f has a relave (local) maximum at (c, f(c)). 3. If f '(x) is posive on both sides of c or negave on both sides of c, then f(c) is neither a relave minimum nor a relave maximum. 11

12 Example 2: Find the relative extrema of the function the interval (0, 2π). in Example 3: Find the relative extrema of f(x) = (x 2 4) 2/3 12

13 Example 4: Find the relative extrema of ***Be sure to consider the domain of this function (and other functions). The function is not defined at x=0. Therefore this value needs to be used with the critical points to determine the test intervals. 13