Maximum and Minimum Values and Sketch Graphs 2

Maximum and Minimum Values and Sketch Graphs 2. 1 OBJECTIVE • • Find relative extrema of a continuous function using the First-Derivative Test. Sketch graphs of continuous functions. Copyright © 2014 Pearson Education, Inc.

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs DEFINITIONS: A function f is increasing over I if, for every a and b in I, if a < b, then f (a) < f (b). (If the input a is less than the input b, then the output for a is less than the output for b. A function f is decreasing over I if, for every a and b in I, if a < b, then f (a) > f (b). (If the input a is less than the input b, then the output for a is greater than the output for b. ) Copyright © 2014 Pearson Education, Inc. Slide 2 - 2

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 1 If f (x) > 0 for all x in an interval I, then f is increasing over I. If f (x) < 0 for all x in an interval I, then f is decreasing over I. Copyright © 2014 Pearson Education, Inc. Slide 2 - 3

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs DEFINITION: A critical value of a function f is any number c in the domain of f for which the tangent line at (c, f (c)) is horizontal or for which the derivative does not exist. That is, c is a critical value if f (c) exists and f (c) = 0 or f (c) does not exist. Copyright © 2014 Pearson Education, Inc. Slide 2 - 4

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs DEFINITIONS: Let I be the domain of f : f (c) is a relative minimum if there exists within I an open interval I 1 containing c such that f (c) ≤ f (x) for all x in I 1; and f (c) is a relative maximum if there exists within I an open interval I 2 containing c such that f (c) ≥ f (x) for all x in I 2. Copyright © 2014 Pearson Education, Inc. Slide 2 - 5

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 2 If a function f has a relative extreme value f (c) on an open interval; then c is a critical value. So, f (c) = 0 or f (c) does not exist. Copyright © 2014 Pearson Education, Inc. Slide 2 - 6

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 3: The First-Derivative Test for Relative Extrema For any continuous function f that has exactly one critical value c in an open interval (a, b); F 1. f has a relative minimum at c if f (x) < 0 on (a, c) and f (x) > 0 on (c, b). That is, f is decreasing to the left of c and increasing to the right of c. Copyright © 2014 Pearson Education, Inc. Slide 2 - 7

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THEOREM 3: The First-Derivative Test for Relative Extrema (continued) F 2. f has a relative maximum at c if f (x) > 0 on (a, c) and f (x) < 0 on (c, b). That is, f is increasing to the left of c and decreasing to the right of c. F 3. f has neither a relative maximum nor a relative minimum at c if f (x) has the same sign on (a, c) and (c, b). Copyright © 2014 Pearson Education, Inc. Slide 2 - 8

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Graph over the f (c) Sign of f (x) Increasing or Sketch Graphs interval (a, b) for x in (a, c) for x in (c, b) decreasing Relative minimum – + Decreasing on (a, c]; increasing on [c, b) Relative maximum + – Increasing on (a, c]; decreasing on [c, b) Copyright © 2014 Pearson Education, Inc. Slide 2 - 9

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Graph over the f (c) Sign of f (x) Increasing or Sketch Graphs interval (a, b) for x in (a, c) for x in (c, b) No relative maxima or minima – – decreasing Decreasing on (a, b) Increasing on (a, b) + Copyright © 2014 Pearson Education, Inc. + Slide 2 - 10

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1: Graph the function f given by and find the relative extrema. Suppose that we are trying to graph this function but do not know any calculus. What can we do? We can plot a few points to determine in which direction the graph seems to be turning. Let’s pick some x-values and see what happens. Copyright © 2014 Pearson Education, Inc. Slide 2 - 11

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (continued): Copyright © 2014 Pearson Education, Inc. Slide 2 - 12

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Example 1 (continued): Sketch Graphs We can see some features of the graph from the sketch. Now we will calculate the coordinates of these features precisely. 1 st find a general expression for the derivative. 2 nd determine where f (x) does not exist or where f (x) = 0. (Since f (x) is a polynomial, there is no value where f (x) does not exist. So, the only possibilities for critical values are where f (x) = 0. ) Copyright © 2014 Pearson Education, Inc. Slide 2 - 13

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (continued): These two critical values partition the number line into 3 intervals: A (– ∞, – 1), B (– 1, 2), and C (2, ∞). Copyright © 2014 Pearson Education, Inc. Slide 2 - 14

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (continued): 3 rd analyze the sign of f (x) in each interval. Test Value x = – 2 x=0 x=4 Sign of f (x) + – + Result f is increasing f is decreasing on on (–∞, – 1] [– 1, 2] Copyright © 2014 Pearson Education, Inc. f is increasing on [2, ∞) Slide 2 - 15

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 1 (concluded): Therefore, by the First-Derivative Test, f has a relative maximum at x = – 1 given by Thus, (– 1, 19) is a relative maximum. And f has a relative minimum at x = 2 given by Thus, (2, – 8) is a relative minimum. Copyright © 2014 Pearson Education, Inc. Slide 2 - 16

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 1 Graph the function relative extrema. given by , and find the Relative Maximum at: Relative Minimum at: Copyright © 2014 Pearson Education, Inc. Slide 2 - 17

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 2: Find the relative extrema for the Function f (x) given by Then sketch the graph. 1 st find f (x). Copyright © 2014 Pearson Education, Inc. Slide 2 - 18

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 2 (continued): Then, find where f (x) does not exist or where f (x) = 0. Note that f (x) does not exist where the denominator equals 0. Since the denominator equals 0 when x = 2, x = 2 is a critical value. f (x) = 0 when the numerator equals 0. Since 2 ≠ 0, f (x) = 0 has no solution. Thus, x = 2 is the only critical value. Copyright © 2014 Pearson Education, Inc. Slide 2 - 19

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 2 (continued): 3 rd x = 2 partitions the number line into 2 intervals: A (– ∞, 2) and B (2, ∞). So, analyze the signs of f (x) in both intervals. Test Value x=0 x=3 Sign of f (x) – + Result f is decreasing on (– ∞, 2] Copyright © 2014 Pearson Education, Inc. f is increasing on [2, ∞) Slide 2 - 20

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Example 2 (continued): Therefore, by the First-Derivative Test, f has a relative minimum at x = 2 given by Thus, (2, 1) is a relative minimum. Copyright © 2014 Pearson Education, Inc. Slide 2 - 21

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Example. Graphs 2 (concluded): Sketch We use the information obtained to sketch the graph below, plotting other function values as needed. Copyright © 2014 Pearson Education, Inc. Slide 2 - 22

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 2 Find the relative extrema of the function Then sketch the graph. First find given by : Copyright © 2014 Pearson Education, Inc. Slide 2 - 23 .

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 2 Continued Second find where does not exist or where So and Thus when and . . are the critical values. h’(x) exists everywere. Copyright © 2014 Pearson Education, Inc. Slide 2 - 24

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Quick Check 2 Continued Sketch Graphs Third, and partitions the number line into three intervals: A , B , and C. So analyze the signs of in all three intervals. Interval Test Value Sign of Result Thus, there is a minimum at . Therefore, Copyright © 2014 Pearson Education, Inc. is a minimum. Slide 2 - 25

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 2 Concluded From the information we have gathered, the graph of Copyright © 2014 Pearson Education, Inc. Slide 2 - 26 looks like:

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 3 Find the relative extrema of the function Then sketch the graph. First find given by : Copyright © 2014 Pearson Education, Inc. Slide 2 - 27 .

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 3 Continued Second, find where does not exist or where . Note: does not exist when the denominator = 0. So not exist when. Also, there is no value of that makes. Thus there is a critical value at . Copyright © 2014 Pearson Education, Inc. Slide 2 - 28 does

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 3 Continued Third, and partitions the number line into two intervals: . So analyze the signs of for both intervals. Interval Test Value Sign of Results Thus there is no extrema for . Copyright © 2014 Pearson Education, Inc. Slide 2 - 29

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Quick Check 3 Concluded Using the information gathered, the graph of Copyright © 2014 Pearson Education, Inc. looks like: Slide 2 - 30

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Section Summary • A function that is increasing over an interval. if, for all • A function such that is decreasing over an interval. if, for all Copyright © 2014 Pearson Education, Inc. Slide 2 - 31 and such in

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Section Summary Continued • Using the first derivative, a function is increasing over an open interval if, for all in , the slope of the tangent line at is positive; that is, . Similarly, a function is decreasing over an open interval if, for all in , the slope of the tangent line is negative; that is. • A critical value is a number in the domain of such that or does not exist. The point is called a critical point. • A relative maximum point is higher than all other points in some interval containing it. Similarly, a relative minimum point is lower than all other points in some interval containing it. The y-value of such a point is called a relative maximum (or minimum) value of the function. Copyright © 2014 Pearson Education, Inc. Slide 2 - 32

2. 1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs Section Summary Concluded • Minimum and maximum points are collectively called extrema. • Critical values are candidates for possible relative extrema. The First-Derivative Test is used to classify a critical value as a relative minimum, a relative maximum, or neither. Copyright © 2014 Pearson Education, Inc. Slide 2 - 33