3.3 Increasing and Decreasing Functions and the First Derivative Test

1. 3.3 Increasing and Decreasing Functions and the First Derivative Test

2. After this lesson, you should be able to: • To determine algebraically when a function is increasing, decreasing. • Apply the First Derivative Test to find relative extrema of a function.

3. a b c

4. a b c Theorem 3.5 Test for Increasing and Decreasing Functions Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). • If f’(x)>0 for all x in (a, b), then f is increasing on [a, b]. • If f’(x)<0 for all x in (a, b), then f is decreasing on [a, b]. • If f’(x)=0 for all x in (a, b), then f is constant on [a, b].

5. Will the signs always alternate at two side of each Critical Number? Example? Example Example 1 Find the open interval(s) on which the function is increasing or decreasing. Solution Note that f(x) is differentiable on the entire real number set. Set f’(x)= 0 to find the critical number(s). Then the zeros of f ’(x) are x = –1 and x = 1

6. Example Example 2 Sketch the graph of f(x) if: What are the critical numbers? Which critical numbers are extrema? Why? (.2*x^5-.5*x^4-5x^3+18x^2-60)/30 + – + + –4 3 0

7. Guidelines for Finding Intervals on Which a Function Is Increasing or Decreasing Let f(x) be continuous on the interval (a, b) • 1. Locate the critical numbers of f in (a, b), and use these numbers to determine test intervals. • Determine the sign of f’(x) at one test value in each of the intervals. • Use Theorem 3.5 to determine whether f is increasing or decreasing on each interval.

9. Example Example 2 Sketch the graph of f(x) if: What are the critical numbers? Which critical numbers are extrema? Why? (.2*x^5-.5*x^4-5x^3+18x^2-60)/30 + – + + –4 3 0

10. Theorem 3.6 1st Derivative Test Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows. • If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f. • If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f. • If f’(x) does not change sign at c, then f(c) is neither a relative minimum nor a relative maximum.

11. Monotone A function that is only increasing or only decreasing over its domain is called strictly monotonic. Monotone increasing: always increasing Example: Monotone decreasing: always decreasing Example: Note: The function is not monotone in the domain.

12. Properties of the graph of a function f

13. Curve Sketching - 8 Part Procedure Example 3 1. Domain: all x  R 2. 1st Derivative: – + 3

14. Curve Sketching - 8 Part Procedure 3. 2nd Derivative: + + – –

15. Curve Sketching - 8 Part Procedure 4. Limits of Special Interest: (see the domain) + + *When taking the limit of a polynomial function as x approaches (positive or negative) infinity, the term with the highest power is all that is needed to be considered.

16. Curve Sketching - 8 Part Procedure 5. Summary: Draw domain 3 – + Look at 1st Derivative 3 + + + Look at 2nd Derivative 3

17. Curve Sketching- 8 Part Procedure 6. Points: (3, 1) and (0, 10)

18. Curve Sketching - 8 Part Procedure 7. Graph: 8. Range, etc: Vertex: (3, 1) Range: Axis of Symmetry: x = 3 Min: y = 1 when x = 3 No max Increasing on Parabola Decreasing on

19. Example Example 4 Find the relative extrema of the function f(x) in the interval [0, 2]: Solution Note that f(x) is differentiable on [0, 2]. Set f’(x)= 0 to find the critical number(s). Then the zeros of f ’(x) are x = /6 and x = 5/6

20. Example Example 5 Find the relative extrema of the function f(x): ((x^2-4)^2)^(1/3) Solution Set f’(x)= 0 to find the critical number(s). Then the zeros of f ’(x) are x = 0 and f’(x) does not exist at x =±2.

21. Example Example 6 Find the relative extrema of the function f(x): Solution Set f’(x)= 0 to find the critical number(s). Then the zeros of f ’(x) are x = ±1 and f’(x) does not at x = 0

22. Homework Section 3.3 page 181 #3-13 odd, 17, 18-23 all (full curve sketches), 25-29 odd