Aim: How can we classify relative extrema as either relative minimums or relative maximums?

1. Aim: How can we classify relative extrema as either relative minimums or relative maximums? Do Now: • The height of a ball t seconds after it is thrown upward from a height of 32 feet and with an initial velocity of 48 feet per second. • Verify that f(1) = f(2) • According to Rolle’s Theorem, what must be the velocity at some time in the interval [1, 2]?

2. Increasing and Decreasing Functions x = a x = b as x moves to the right Increasing Decreasing Constant f’(x) = 0 f’(x) > 0 f’(x) < 0 Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. If f’(x) > 0, for all x in (a, b), then f is increasing on [a, b] Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1.

3. Increasing and Decreasing Functions x = a x = b Increasing Decreasing Constant f’(x) = 0 f’(x) > 0 f’(x) < 0 Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 2. If f’(x) < 0, for all x in (a, b), then f is decreasing on [a, b]

4. Increasing and Decreasing Functions x = a x = b Increasing Decreasing Constant f’(x) = 0 f’(x) > 0 f’(x) < 0 Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 3. If f’(x) = 0, for all x in (a, b), then f is constant on [a, b]

5. Increasing (0, 0) (1, -.5) Decreasing Increasing Model Problem Find the open intervals on which is increasing or decreasing. Find critical numbers = 0 x = 0, 1 no points where function is undefined

6. Increasing (0, 0) (1, -.5) Decreasing Increasing Model Problem Find the open intervals on which is increasing or decreasing. - < x < 0 (-, 0) 0 < x < 1 (0, 1) 1 < x <  (1, ) x = -1 x = ½ x = 2 f’(1/2) = -3/4 f’(-1) = 6 f’(2) = 6 Increasing Decreasing Increasing

7. Guidelines To Find Intervals on Which a Function is Increasing or Decreasing Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing • Located the critical numbers of f in (a, b) and use to determine test intervals. 2. Determine sign of f’(x) at one test value in each interval 3. Determine status on each interval.

8. The First 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. 1. If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f. ( – ) ( + ) f’(x) < 0 f’(x) > 0 b a c

9. The First 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, the f(c) can be classified as follows. 2. If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f. ( + ) ( – ) f’(x) > 0 f’(x) < 0 b a c

10. 0 2π Model Problem 1 Find the relative extrema of the function in the interval (0, 2) continuous, differentiable, no where undefined 1. Find critical values

11. Model Problem 1 Find the relative extrema of the function in the interval (0, 2) 2. Create table of intervals

12. Model Problem 1 Find the relative extrema of the function in the interval (0, 2). 3. Conclusion

13. undefined at zeros of denom. x = ±2 Model Problem 2 Find the relative extrema of 1. Find critical values continuous, differentiable at all but ±2 Critical values x = 0

14. Model Problem 2 Find the relative extrema of 2. Create table of intervals

15. f has a relative minimum at (-2, 0) & (2, 0) f has a relative maximum at Model Problem 2 Find the relative extrema of 3. Conclusion

16. Aim: How can we classify relative extrema as either relative minimums or relative maximums? Do Now: Find the value or values of c that satisfy for the function on the interval [3, 9].

17. Model Problem 3 Find the relative extrema of the function = x2 + x-2 1. Find critical values f’(x) = 0 @ ±1; f’(0) is undefined

18. Model Problem 3 Find the relative extrema of the function = x2 – x-2 2. Create table of intervals

19. Model Problem 3 Find the relative extrema of the function = x2 – x-2 3. Conclusion f has a relative minimum at (-1, 2) & (1, 2)

20. Model Problem 4 Neglecting air resistance, the path of a projectile that is propelled at an angle  is where y is the height, x is the horizontal distance, g is the acceleration due to gravity, v0 is the initial velocity, and h is the initial height. Let g = -32 feet per second per second, v0 = 24 feet per second, and h = 9 feet. What value of  will produce the maximum horizontal distance?

21. Model Problem 4 use values supplied & Quad. Form.