Section MWF 12 1pm SR 117

Transcription

1 Math 1431 Section 1485 MWF 1 1pm SR 117 Dr. Melahat Almus almus@math.uh.edu COURSE WEBSITE: Visit my website regularly for announcements and course material! If you me, please mention your course (1431) in the subject line. Check your CASA account for quiz due dates; don t miss any quizzes. BUBBLE IN PS ID VERY CAREFULLY! If you make a bubbling mistake, your scantron will not be saved in the system and you will not get credit for it even if you turned it in. Bubble in Popper Number. Be considerate of others in class. Respect your friends and do not distract anyone during the lecture. 1

2 POPPER # Question# Find the interval(s) over which the function is increasing: 3 f x x 6x 9x a) (, 3) b) ( 3, 1) c) (, 3) and ( 1, ) d) ( 1, ) e) None of these

3 Section 3.4 Extreme Values Local Extreme Values Definition: Suppose that f is a function defined on open interval I and c is an interior point of I. The function f has a local minimum at x c if f c f x for all x in I (that is, for all x sufficiently close to c ). The function f has a local maximum at x c if f c f x for all x in I (that is, for all x sufficiently close to c ). In general, if f has a local minimum or maximum at x local extreme value of f. c, we say that f c is a When the graph of a function is given, we can easily find the local extreme values by inspection. 3

4 This graph suggests that local maxima or minima occur at the points where the tangent line is horizontal or where the function is not differentiable and this is true. Fact: Suppose that the function f is defined on an open interval containing the number c. If f has a local minimum or maximum at x c, then f' c 0 or f ' c does not exist. This fact gives us one of the tools for finding local extreme values of a function defined by a formula. Definition: The interior points c of the domain of a function f for which f' c 0 or f ' c does not exist are called critical points for f. Example 1: Find the critical points for the function: f x x 4x. 4

5 Example : Find the critical points for the function: f x x. x 1 13 / Example 3: Find the critical points for the function: f x x. 5

6 Finding extreme values Theorem: The First-Derivative Test Suppose that c is a critical point for f and f is continuous at c. If there is a positive number r such that: (i) f' x 0 for x in c r,c and f' x 0 for x in c,c r, then a local minimum. (ii) f' x 0 for x in c r,c and f' x 0 for x in c,c r, then (iii) a local maximum. f ' x has the same sign for all x in c a local extreme value for f. r,c or c,c r, then f c is f c is f c is not 6

7 Example 1: Locate all local extrema: f x x 5x 5x. Given: f' x 5x 4 0x 3 15x 5x x 3 x 1 Example : Find the critical points and local extreme points: Given: f' x x x 8 x 4 f x x. x 4 7

8 Example 3: Find the critical points. Determine the intervals where the function is increasing/decreasing. Locate all local extrema. x 1 f x x 1. Given: f' x 4x 4x x 1x 1x 1 Example 4: Find the critical points and classify them as local min/max or neither: 5 / 3 f x x. 8

9 Example 5: Find the critical points for the function: f x x 1 4. Question# point at x = 0. A sign chart is shown below for the function f. Classify the critical a. local maximum b. local minimum c. neither Question# point at x =. A sign chart is shown below for the function f. Classify the critical a. local maximum b. local minimum c. neither 9

10 Questions: If f > 0, what conclusion can be made about f? If f < 0, what conclusion can be made about f? If f (c) = 0 and f (c) > 0, what conclusion can be made about f (c)? If f (c) = 0 and f (c) < 0, what conclusion can be made about f (c)? Sometimes it is difficult to study the sign of the derivative function. For some cases, it may be easier to use the following test: Theorem: The Second-Derivative Test Let c be a critical point for f where f' c 0 and f '' c exists. (i) If f'' c 0, then (ii) If f'' c 0, then f c is a local minimum value, f c is a local maximum value, (iii) If f'' c 0, then this test is inconclusive. 10

11 Example 6: Determine whether the critical point x 1 is a local minimum or 4 3 maximum for the function: f x 3x 8x 6x 4x 5. Example 7: Given f''x 6x 1 and the critical points of x=4. Classify these critical points as local min/max. f x are x=1 and 11

12 POPPER # Question# What are the critical points for f x Given: f' x x x 1x 1 x x 1? a) x 1, x 1 b) x 1, x 1, x 0 c) x 0 d) x 1, x 0 e) No critical points Question# Find all critical points: f x 9 x 3 5 Given: f' x 6x 59 x / 5 a) x 3 b) x 3, x 3 c) x 0 d) x 3, x 3, x 0 e) No critical points 1