Calculus and Analytical Geometry

1. MTH 104 Calculus and Analytical Geometry Lecture # 12

2. Increasing and decreasing functions • The terms increasing, decreasing, and constant are used to describe the behavior of a function over an interval as we travel left to right along its graph.

3. Increasing and decreasing functions • Let f be defined on an interval, and let x1 and x2 denote points in that interval. • f is increasing on the interval if f(x1) < f(x2) whenever x1 < x2. • f is decreasing on the interval if f(x1) > f(x2) whenever x1 < x2. • f is constant on the interval if f(x1) = f(x2) for all points x1 and x2 .

4. Increasing/decreasing function

5. Increasing/decreasing test • Let f be a function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b). • If for every value of x in (a, b), then f is increasing on [a, b]. • If for every value of x in (a, b), then f is decreasing on [a, b]. • If for every value of x in (a, b), then f is constant on [a, b]

6. Sign Analysis to Determine where f (x) is Increasing/Decreasing 1. Find all values of x for which is discontinuous and identify open intervals with these points. 2. Test a point c in each interval to check the sign of a. If fis increasing on that interval. b. If fis decreasing on that interval. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

7. Example solution Find the intervals on which is increasing and the intervals on which it is decreasing. Differentiating with respect to x This implies that f is decreasing on This implies that f is increasing on

8. Example solution Find the intervals on which is increasing and intervals on which it is decreasing. f is increasing on f is increasing on Thus f is increasing on

9. Example Use the graph below to make a conjecture about the intervals on which f is increasing or decreasing.

10. Use increasing/decreasing test to verify your conjecture.

13. Concavity • Two ways to characterize the concavity of a differentiable function f on an open onterval. • f is concave up on an open interval if its tangent lines have increasing slopes on that interval and is concave down if they have decreasing slopes. • f is concave up on an open interval if its graph lies above its tangent lines on that interval and is concave down if it lies below its tangent lines.

14. Concavity Definition If f is differentiable on an open interval I, then f is said to be concave up on I if f’ is increasing on I, and f is said to be concave down on I if f’ is decreasing on I Test for concavity Let f be twice differentiable on a open interval I. If for every value of x in I, then f is concave up on I. If for every value of x in I, then if is concave down on I.

15. Inflection Points If f is continuous on an open interval containing a value x0 , and if f changes the direction of its concavity at the point (x0 , f(x0)), then we say that f has an inflection point at x0, and we call the point an inflection point of f.

16. Inflection Point To find inflection points, find any point, c, in the domain where is undefined. If changes sign from the left to the right of c, Then (c, f(c))is an inflection point of f.

17. Example Given , find the intervals on which f is increasing/decreasing and concave up/down. Locate all points of inflection. Solution at x=0 and at x=2 0 2

18. Sign analysis(test for increasing/decreasing function) of Sign analysis of 1

19. Point of inflection Since f changes from concave down to concave up at x=1, so there is an inflection point at x=1. Inflection point is (1, f(1))=(1,-1).

20. Example • Given • Find where this function is increasing/decreasing. • Find where this function is concave up/down. Solution We see that at x=1 1 4 -3 Test point f is increasing on f is decreasing on

21. 2 5 0 Test point f is concave down on f is concave down on Inflection point at x=2

22. Example Given on the interval Use first and second derivatives of f to determine where f is increasing, decreasing, concave up, and concave down. Locate all points of inflection. solution put

23. 0 And is only solution which lies in the interval

24. 0 There is a point of inflection at x= since f changes from concave down to concave up at that point.

25. Inflection Points Inflection points mark the places on the curve y = f(x) where the rate of change of y with respect to x changes from increasing to decreasing, or vice versa.