1. Introduction to Graph Theory

2. Critical Points (x-value)

3. Critical Points (x-value)
If f’(c) = 0 or is undefined, then c is a critical point of f(x)

4. Critical Points (x-value)
If f’(c) = 0 or is undefined, then c is a critical point of f(x)
(f”(c) = 0 is also called a stationary point)

5. Extrema
Absolute (global) Extrema: the max/min value of a function…there can be only one abs. max/min (output), but it may occur at more than one x-value (input).

6. Extrema
Absolute (global) Extrema: the max/min value of a function…there can be only one abs. max/min (output), but it may occur at more than one x-value (input). Relative (local) Extrema: Occur at critical points at the top of hills (relative max) or the bottom of valleys (relative min). There may be more than one relative max/mins.

7. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval.

8. The Extreme Value Theorem
If f(x) is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval. It is important to note that if f(x) is not continuous or on a closed interval, then it may or may not have absolute extrema.

9. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval. Absolute extrema occur at end points or critical points.

10. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval. Absolute extrema occur at end points or critical points. I. Examples of finding extrema graphically

11. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval.
II. Steps for finding extrema algebraically on a closed interval
Find the critical points of f on (a, b)

12. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval.
Steps for finding extrema on a closed interval
Find the critical points of f on (a, b)
Evaluate f at each critical point

13. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval.
Steps for finding extrema on a closed interval
Find the critical points of f on (a, b)
Evaluate f at each critical point
Evaluate f at the end points

14. The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on the interval.
Steps for finding extrema on a closed interval
Find the critical points of f on (a, b)
Evaluate f at each critical point
Evaluate f at the end points
Determine the min and max of these output values

15. What f’ says about f…
f’ is positive ↔ f is increasing

16. What f’ says about f…
f’ is positive ↔ f is increasing f’ is negative ↔

17. What f’ says about f…
f’ is positive ↔ f is increasing f’ is negative ↔ f is decreasing

18. What f’ says about f… f’ is positive ↔ f is increasing
f’ is negative ↔ f is decreasing
If f’(c) = 0 or is undefined, then c is a critical point of f(x)
(f”(c) = 0 is also called a stationary point)

19. Increasing/Decreasing & Local Extrema

20. Increasing/Decreasing & Local Extrema
Process
Find f’(x).

21. Increasing/Decreasing & Local Extrema
Process
Find f’(x).
Find critical points, f’(x) = 0 or undefined.

22. Increasing/Decreasing & Local Extrema
Process
Find f’(x).
Find critical points, f’(x) = 0 or undefined.
Create (and label) a Sign Line.

23. Increasing/Decreasing & Local Extrema
Process
Find f’(x).
Find critical points, f’(x) = 0 or undefined.
Create (and label) a Sign Line.
Local max at x = c if f’ changes from positive to negative at x = c

24. Increasing/Decreasing & Local Extrema
Process
Find f’(x).
Find critical points, f’(x) = 0 or undefined.
Create (and label) a Sign Line.
Local max at x = c if f’ changes from positive to negative at x = c
(and c is in the domain of f(x)).

25. Increasing/Decreasing & Local Extrema
Process
Find f’(x).
Find critical points, f’(x) = 0 or undefined.
Create (and label) a Sign Line.
Local max at x = c if f’ changes from positive to negative at x = c.
(and c is in the domain of f(x)).
Local min at x = c if f’ changes from negative to positive at x = c.

26. Concavity: What f ” says about f.

27. Concavity
f " is positive ↔ f is concave up (happy face)

28. Concavity
f " is positive ↔ f is concave up (happy face) f " is negative ↔ f is concave down (sad face)

29. f " is positive ↔ f is concave up (happy face)
Concavity
f " is positive ↔ f is concave up (happy face)
f " is negative ↔ f is concave down (sad face)
If f "(c) = 0 or is undefined, then c is a potential point of inflection

30. f " is positive ↔ f is concave up (happy face)
Concavity
f " is positive ↔ f is concave up (happy face)
f " is negative ↔ f is concave down (sad face)
If f "(c) = 0 or is undefined, then c is a potential point of inflection
To be an actual point of inflection: x=c must be in the domain
of f(x) and f(x) must change concavity at x = c.

31. Concavity f " is positive ↔ f is concave up (happy face)
f " is negative ↔ f is concave down (sad face)
If f "(c) = 0 or is undefined, then c is a potential point of inflection
To be an actual point of inflection: x=c must be in the domain
of f(x) and f(x) must change concavity at x = c.
Process
Find f "(x).
Find PPOI, f "(x) = 0 or undefined.
Create (and label) a Sign Line.