Increasing and Decreasing Functions and the First Derivative Test

1. Increasing and Decreasing Functions and the First Derivative Test Determine the intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema of a function Standard 4.5a

2. y Increasing Decreasing Constant x

3. Test for Increasing and Decreasing Functions Let f be differentiable on the interval (a, b) If f’(x) > 0 then f is increasing on (a, b) If f’(x) < 0 then f is decreasing on (a, b) If f’(x) = 0 then f is constant on (a, b)

4. Definition of Critical Number If f is defined at c, then c is a critical number of f if f’(c)=0 or if f’ is undefined at c.

5. Find the open intervals on which the given function is increasing or decreasing. 1. Find derivative. 2. Set f’(x) = 0 and solve to find the critical numbers. CRITICAL NUMBERS

6. Make table to test the sign f’(x) in each interval. Use the test for increasing/decreasing to decide whether f is increasing or decreasing on each interval.

7. Find the open intervals on which the given function is increasing or decreasing.

8. y Relative maximum Increasing Decreasing Increasing Relative minimum x

9. Definition of Relative Extrema Let f be a function defined at c. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b). f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b). If f(c) is a relative extremum of f, then the relative extremum is said to occur at x = c.

10. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b). relative maximum f(c) f(x) f(x) f(x) f(x) f(x) f(x) f(x) f(x) c

11. 2. f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b). f(x) f(x) f(x) f(x) f(x) f(x) f(c) relative minimum

12. Occurrence of Relative Extrema If f has a relative minimum or a relative maximum when x = c, then c is a critical number of f. That is, either f’(c) = 0 or f’(c) is undefined.

13. First-Derivative Test for Relative Extrema Let f be continuous on the interval (a, b) in which c is the only critical number. On the interval (a, b) if 1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum. 2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum. 3. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum.

14. 1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum. f’(x) is positive f’(x) is negative Relative minimum c

15. 2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum. relative maximum f’(x) is positive f’(x) is negative c

16. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum. Not a relative extremum f’(x) is positive f’(x) is positive c

17. Find all relative extrema of the given function. Find derivative Set = 0 to find critical numbers CRITICAL NUMBERS

18. Relative Maximum (-1, 5) Relative Minimum (1, -3)

19. Find all relative extrema of the given function. Relative max: (-2, 0) Relative min: (0, -2)

20. Find all relative extrema of the given function.

21. Relative max: Relative min:

22. The graph of f is shown. Sketch a graph of the derivative of f.

23. The graph of f is shown. Sketch a graph of the derivative of f.

24. The graph of f is shown. Sketch a graph of the derivative of f.