1. Maximum and Minimum Values
3.1
Maximum and Minimum Values

2. Maximum and Minimum values of a function
Some of the most important applications of calculus are optimization problems, which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function.
So what are maximum and minimum values?
Example:
highest point on the graph of the function f shown is the point (3, 5) so the largest value of f is f (3) = 5.
Likewise, the smallest value is f (6) = 2.
Figure 1

3. Absolute Maximum and Minimum
The maximum and minimum values of f are called extreme values of f.

4. Local Maximum and Minimum
Something is true near c means that it is true on some open interval containing c.

5. Example 1

6. Example 2
Figure 2 shows the graph of a function f with several extrema:
At a : absolute minimum is f(a)
At b : local maximum is f(b) At c : local minimum is f(c) At d : absolute (also local) maximum is f(d)
At e : local minimum is f(e)
Abs min f (a), abs max f (d)
loc min f (c) , f(e), loc max f (b), f (d)
Figure 2

7. Practice!
Define all local and absolute extrema of the graph below

8. Critical Point

9. Counter examples: f’ exists but there is no local min or max
f’ doesn’t exist but there is a local min or max

10. Extreme Value Theorem
The following theorem gives conditions under which a function is guaranteed to have extreme values.

11. Extreme Value Theorem: Absolute min and max
The Extreme Value Theorem is illustrated below:
Note that an extreme value can be taken on more than once.
Figure 7

12. Extreme Value Theorem : Absolute min and max
The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. We start by looking for local extreme values.
Graph of a function f with a local maximum at c and a local minimum at d.
Figure 10

13. Finding Absolute Maximum and Minimum of f:
To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local or it occurs at an endpoint of the interval.
Thus the following three-step procedure always works.

14. Practice
Find the absolute values of the function and where they occur (worksheet 3.1, # 5)

15. 3.2
The Mean Value Theorem

16. Rolle’s Theorem

17. Examples: Figure 1 shows the graphs of four such functions. (a) (b)
(d)
Figure 1

18. The Mean Value Theorem
This theorem is an extension of Rolle’s Theorem

19. Example Show that f(x) satisfies the Mean Value Theorem on [a,b]
f (x) = x3 – x, Interval: a = 0, b = 2.
Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 2] and differentiable on (0, 2).
Therefore, by the Mean Value Theorem, there is a number c in (0, 2) such that
f (2) – f (0) = f (c)(2 – 0)

20. Example - proof f (2) = 6, f (0) = 0, and f (x) = 3x2 – 1,
so this equation becomes:
6 = (3c2 – 1)2
= 6c2 – 2
which gives that is, c = But c must lie in
(0, 2), so

21. The Mean Value Theorem
The Mean Value Theorem can be used to establish some of the basic facts of differential calculus.