1. Chapter 5 Applications of the Derivative Sections 5. 1, 5. 2, 5
Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4

2. Applications of the Derivative
Maxima and Minima
Applications of Maxima and Minima
The Second Derivative - Analyzing Graphs

3. Absolute Extrema Let f be a function defined on a domain D
Absolute Maximum
Absolute Minimum

4. Absolute Extrema
A function f has an absolute (global) maximum at x = c if f (x)  f (c) for all x in the domain D of f.
The number f (c) is called the absolute maximum value of f in D
Absolute Maximum

5. Absolute Extrema
A function f has an absolute (global) minimum at x = c if f (c)  f (x) for all x in the domain D of f.
The number f (c) is called the absolute minimum value of f in D
Absolute Minimum

6. Generic Example

7. Generic Example

8. Generic Example

9. Relative Extrema
A function f has a relative (local) maximum at x  c if there exists an open interval (r, s) containing c such that f (x)  f (c) for all r  x  s.
Relative Maxima

10. Relative Extrema
A function f has a relative (local) minimum at x  c if there exists an open interval (r, s) containing c such that f (c)  f (x) for all r  x  s.
Relative Minima

11. Generic Example
The corresponding values of x are called Critical Points of f

12. Critical Points of f (stationary point) (singular point)
A critical number of a function f is a number c in the domain of f such that
(stationary point)
(singular point)

13. Candidates for Relative Extrema
Stationary points: any x such that x is in the domain of f and f ' (x)  0.
Singular points: any x such that x is in the domain of f and f ' (x)  undefined
Remark: notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.

14. Fermat’s Theorem
If a function f has a local maximum or minimum at c, then c is a critical number of f
Notice that the theorem does not say that at every critical number the function has a local maximum or local minimum

15. Generic Example Two critical points of f that do
not correspond to local extrema

16. Example Find all the critical numbers of Stationary points:
Singular points:

17. Graph of
Local max.
Local min.

18. Extreme Value Theorem
If a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint.
a b
a b
a b
Attains max. and min.
Attains min. but no max.
No min. and no max.
Open Interval
Not continuous

19. Finding absolute extrema on [a , b]
Find all critical numbers for f (x) in (a , b).
Evaluate f (x) for all critical numbers in (a , b).
Evaluate f (x) for the endpoints a and b of the interval [a , b].
The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a , b].

20. Example Find the absolute extrema of
Critical values of f inside the interval (-1/2,3) are x = 0, 2
Absolute Max.
Absolute Min.
Evaluate
Absolute Max.

21. Example Find the absolute extrema of
Critical values of f inside the interval (-1/2,3) are x = 0, 2
Absolute Max.
Absolute Min.

22. Example Find the absolute extrema of
Critical values of f inside the interval (-1/2,1) is x = 0 only
Absolute Max.
Evaluate
Absolute Min.

23. Example Find the absolute extrema of
Critical values of f inside the interval (-1/2,1) is x = 0 only
Absolute Max.
Absolute Min.

24. Increasing/Decreasing/Constant

25. Increasing/Decreasing/Constant

26. Increasing/Decreasing/Constant

27. The First Derivative Test

28. Generic Example
A similar
Observation
Applies at a
Local Max.

29. The First Derivative Test
Determine the sign of the derivative of f to
the left and right of the critical point.
left
right
conclusion
f (c) is a relative maximum
f (c) is a relative minimum
No change
No relative extremum

30. Relative Extrema Example: Find all the relative extrema of
Stationary points:
Singular points: None

31. The First Derivative Test
Find all the relative extrema of
Relative max. f (0) = 1
Relative min. f (4) = -31

32. The First Derivative Test

33. The First Derivative Test

34. Another Example Find all the relative extrema of Stationary points:
Singular points:

35. Stationary points: Singular points: Relative max. Relative min.
ND ND ND +

36. Graph of
Local max.
Local min.

37. Domain Not a Closed Interval
Example: Find the absolute extrema of
Notice that the interval is not closed. Look graphically:
Absolute Max.
(3, 1)

38. Optimization Problems
Identify the unknown(s). Draw and label a diagram as needed.
Identify the objective function. The quantity to be minimized or maximized.
Identify the constraints.
4. State the optimization problem.
5. Eliminate extra variables.
6. Find the absolute maximum (minimum) of the objective function.

39. Optimization - Examples
An open box is formed by cutting identical squares from each corner of a 4 in. by 4 in. sheet of paper. Find the dimensions of the box that will yield the maximum volume. x
4 – 2x x x x
4 – 2x

40. Critical points:
The dimensions are 8/3 in. by 8/3 in. by 2/3 in. giving a maximum box volume of V  4.74 in3.

41. Optimization - Examples
An metal can with volume 60 in3 is to be constructed in the shape of a right circular cylinder. If the cost of the material for the side is $0.05/in.2 and the cost of the material for the top and bottom is $0.03/in.2 Find the dimensions of the can that will minimize the cost.
top and bottom
cost
side

42. So
Sub. in for h

43. Graph of cost function to verify absolute minimum:
2.5
So with a radius ≈ 2.52 in. and height ≈ 3.02 in. the cost is minimized at ≈ $3.58.

44. Second Derivative

45. Second Derivative - Example

46. Second Derivative

47. Second Derivative

48. Concavity Let f be a differentiable function on (a, b).
1. f is concave upward on (a, b) if f ' is increasing on aa(a, b). That is f ''(x)  0 for each value of x in (a, b).
2. f is concave downward on (a, b) if f ' is decreasing on (a, b). That is f ''(x)  0 for each value of x in (a, b).
concave upward
concave downward

49. Inflection Point
A point on the graph of f at which f is continuous and concavity changes is called an inflection point.
To search for inflection points, find any point, c in the domain where f ''(x)  0 or f ''(x) is undefined.
If f '' changes sign from the left to the right of c, then (c, f (c)) is an inflection point of f.

50. Example: Inflection Points
Find all inflection points of

51. Inflection point at x  2 2

53. The Point of Diminishing Returns
If the function represents the total sales of a particular object, t months after being introduced, find the point of diminishing returns.
S concave up on
S concave down on
The point of diminishing returns is at 20 months (the rate at which units are sold starts to drop).

54. The Point of Diminishing Returns
S concave down on
Inflection point
S concave up on