1. Derivative and properties of functions
Outline
Maximum and minimum
Fermat’s theorem
The closed interval method

2. Maximum and minimum
Definition. A function f has an absolute maximum (or global maximum) at c if
where D is the domain of f. The number f(c) is called the maximum value. Similarly, we can define absolute minimum (or global minimum) and minimum value.
The maximum and minimum values are called extreme values of f.
Definition. A function f has a local maximum (or relative maximum) at c if Similarly, we can define local minimum (or relative minimum).

3. Remark Both the absolute maximum value and the absolute
minimum value are unique. But the absolute maximum
point or minimum point may not be unique.
When the domain is a closed interval, the endpoint can
NOT be a local maximum point or minimum point.

4. Fermat’s theorem
The extreme value theorem If f is continuous on [a,b], then f attains its extreme values.
Fermat’s theorem If f has a local maximum or minimum at c, and if exists, then
Proof. Suppose f has local maximum.

5. Remark
is only a necessary condition but not sufficient. That is, if c may not be a maximum or minimum point of f.
A typical example is f(x)=x3. At x=0, but f has no maximum or minimum at 0.
If f has maximum or minimum at c, may not exist. For example, f(x)=|x| at x=0.

6. Critical number
Definition A critical number of a function f is a number c in the domain of f such that or does not exist.
Fermat’s theorem is: If f has a local maximum or minimum at c, then c is a critical number of f.
Ex. Find all critical numbers of
Sol.
So the critical numbers are 3/2 and 0.

7. The closed interval method
To find the absolute maximum and minimum values of a
continuous function f on a closed interval [a,b]:
Find the values of f at all critical numbers.
Find the values of f at endpoints.
The largest of all the above values is the global maximum value and the smallest is the global minimum value.

8. Example Ex. Find the extreme values of on [-1,2]. Sol.
So the absolute maximum value is f(2)=4 and the absolute minimum value is f(-1)=–8.

9. Example Ex. Find the extreme values of on [0,1]. Sol.
No critical numbers!
Absolute maximum value absolute minimum value

10. Mean value theorem Outline Rolle’s theorem
Lagrange’s mean value theorem

11. Rolle’s theorem Rolle’s Theorem Let f be a function that satisfies the
following hypotheses:
1. f is continuous on the closed interval [a,b]. ( )
2. f is differentiable in the open interval (a,b). ( )
3. f(a)=f(b).
Then there exists a number c in (a,b) such that

12. Example Ex. Prove that the equation has exactly one root.
Sol. Since f(0)<0, f(1)>0, by the intermediate theorem, there
exists a root. On the other hand, suppose there are two roots,
f(a)=f(b)=0, then by Rolle’s theorem, there is a c such that
But, this is a contraction.

13. Example Ex. Suppose and Prove that there is a number such that
Analysis To use Rolle’s theorem, we need to find a function
F, such that since
Does the F exist? No.
Can we change into
Sol. Let By the given condition, we have
Then by Rolle’s
theorem, such that and hence

14. Question Suppose exists on [1,2], f(1)=f(2)=0,
Prove that there is a number such that
Sol.

15. Question Q1. Suppose and Prove that for any such that Sol.
Q2. Suppose Let k be a
positive integer. Prove that such that

16. Question Q1 Suppose f has second derivative on [0,1] and f(0)=f(1)=0.
Prove that such that
Sol.

17. Lagrange’s mean value theorem
Theorem Let and Then there is a
number c in (a,b) such that
Proof. Let
Then and F(a)=F(b)=f(a). By Rolle’s
theorem, there is a number c in (a,b) such that or,

18. Applications
Corollary If for all x in an interval (a,b), then f is constant in (a,b).
Proof.
Corollary If for all x in (a,b), then is
constant, that is, f(x)=g(x)+c where c is a constant.

19. Example Ex. Prove the identity Sol. Let then
Ex. Suppose prove that where c is a
constant.
Sol.

20. Example Ex. Prove the inequality
Sol. The inequality is equivalent to or
Let f(x)=lnx. By Lagrange’s mean value theorem,
where hence
Therefore the inequality follows.

21. Question
Prove (1) (2)
Sol. (1)
(2)

22. Question
Prove when b>a>e,
Sol.

23. Homework 8
Section 4.1: 53, 54, 55, 63, 74, 75
Section 4.2: 5, 18, 20, 25, 27, 28, 29, 30, 36