1. §4.3. How f   f  affect shape of f
Recall: the fundamental goal of differential
calculus is to find the information on f using
information from its derivative functions.
Topics:
Information from f 
Information from f 

2. I. Information from f  To tell where f is increasing or decreasing,
all we need is the information from f .

3. Increasing and decreasing test:
If f (x)>0 on (a,b),
then f is increasing ()
on (a,b)
(2) If f (x)<0 on (a,b),
then f is decreasing ()

4. Question: How to tell where f (x) > 0 or
where f (x) < 0? Notice that intervals on
which f  is positive or negative are separated
by critical points. That is, critical numbers are
endpoints of those intervals. Thus, to see where
f is increasing or decreasing, we should first find
all c.n. And then use c.n. to find where the sign
of derivative changes.

5. Comment: (1) f  doesn’t always change sign at a c.n..
Ex 1: f(x) = x3. f (0) = 0. So x = 0 is the only c.n. of f. But f is always increasing over its domain. So f  does not change its sign at x = 0
(2) Recall: Not every c.n. gives rise to a local extreme value of f . So we need a test to determine whether a c.n. gives a local max or a local min value.

6. First derivative test(FDT) :
Suppose c is a c.n. of a function f
(a) If f  changes from
+ to – at c, then f
has a local max at c
(b) If f  changes from
– to + at c, then f
has a local min at c
(c) If f  does not change
sign at c, then f has
no local extreme at c

16. II. Information from f 
Concavity
Def: (i) If the graph of f
lies above all of its tangents
on an interval I, then it is
said to be concave up on I ()
(ii) If the graph of f lies
below all of its tangents on
an interval I, then it is said
to be concave down on I ()

18. Concavity test:
(i) If f (x) > 0 on interval I,
then graph of f is  on I
(ii) If f (x) < 0 on interval I,
then graph of f is 

19. Comment: A good way to remember concavity test is:
(1) If we can use its curve to hold some water, then f  is positive
(2) If we cannot use its curve to hold any water, then f  is negative

20. Inflection Point (i.p.)
Def: A point P(c,f(c)) on a curve is called an i.p. if the curve changes concavities at P (i.e., from  to , or from  to ). c is called an inflection number (i.n.)

21. (i) To find i.n. of f, we should consider:
Comment:
(i) To find i.n. of f, we should consider:
Case 1: all numbers at which f (x) = 0.
Case 2: all numbers at which f (x) does not exist.
Caution: The numbers in case 1 and the numbers in
case 2 contain all inflection numbers. But not every
number in case 1 or case 2 is an inflection number.

24. Comment (continued):
(ii) After step (i), to decide whether a number is an i.n. of f, it is necessary to decide whether f  changes sign at this number.
(iii) Like c.n., an i.n. c must be in Df .
(iv) Also like c.n., we use all possible i.n. to find where the sign of f  changes.

25. Question: How to tell where f (x) > 0 or
where f (x) < 0? Notice that intervals on
which f  is positive and negative are separated
by inflection numbers. That is, all possible
inflection numbers are endpoints of those
intervals. Thus, to see where f is concave up or
concave down, we should first find all possible
i.n. and then use them to find where sign of f 
changes.