§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 

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

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 ()

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.

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.

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

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 ()

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 

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

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.)

(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.

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.

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.