1. Applications of differentiation
Five main topics:
(Applied) maximum and minimum problems (§4.1, §4.7)
Mean value theorem (§4.2)
Curve sketching (§4.3, §4.5)
Antiderivatives(§4.9)

2. Remember that the major goal of differential
calculus is to discover the properties for the
given function using its derivatives. Up to this
point, we know how to compute the derivative
for a given function. Now we are in a better
position to investigate some of the applications
of derivatives.

3. §4.1. Max and min values Topics: Terminology
Fermat’s theorem and critical points
Extreme values

4. Terminology
Def: (i) f has an absolute maximum (or global maximum,or maximum) at c if f(c)  f(x) for any xDf. f(c) is called the maximum value of f on Df.
(ii) f has an absolute minimum (or global minimum, or minimum) at c if f(c)  f(x) for any xDf. f(c) is called the minimum value of f on Df.
(iii) The maximum and minimum values of f are called the extreme values of f on Df.

5. Def: (i) f has a relative maximum (or local maximum) at x = c if there exists an open interval I containing c such that f(c)  f(x) for any xI.
(ii) f has a relative minimum (or local minimum) at x = c if there exists an open interval I containing c such that f(c)  f(x) for any xI.
(iii) The local maximum and local minimum values are called the local extreme values of f.
Note:The open interval I must be inside Df . Thus if Df is a closed interval, local extreme values cannot occur at the endpoints of Df .

8. II. Fermat’s theorem and critical points
From the last example, we see that f has local extreme values when x = 0, 1, and 3. Also f has horizontal tangent lines at those points when x = 0, 1, and 3. In general, we have
Fermat’s Theorem: If f has a local extreme value at c, and if f (c) exists, then f (c) = 0.

9. Comment: (1) The converse of the theorem is
not true in general. Ex: f(x) = x3. f (0) = 0 but f
has no local extreme value at x = 0.
(2) The condition “if f (c) exists” cannot be
removed. Ex: f(x) = |x| achieves its local min at
x = 0 but f (0) does not exist.
(3) “Fermat’s theorem + comment (2)” implies
that to find local extreme values, we should
locate all points x at which f (x) = 0 and all
points at which f (x) does not exist.

10. Comment (3) leads to the following definition:
Def: c Df is said to be a critical number (c.n.) of f if either f (c) = 0 or f (c)
does not exist. (c,f(c)) is called a critical point (c.p.).
Comment: There two types of c.n..
Type I: f (c) = Type II: f (c) does not exist.
In terms of c.n., we have:
Theorem: If f has a local extreme at c, then c must be a c.n. of f .

11. Comment:
(1) Not every c.n. of f gives rise to a local extreme value.
Ex 2: f(x) = x3. f (x) = 3x2. f has a c.n. 0. But f has no local extreme value at 0.
(2) A c.n. c must be in the domain of f
Ex 3: f(x) = 1/(x – 1). f (x) = – 1/(x – 1)2.
f (1) doesn’t exist. But x = 1 is not a c.n. since 1 is not in Df .

15. III. Extreme values
Extreme Value Theorem (EVT): If f is continuous on closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b].

18. Comment: EVT says that a continuous function on a closed interval has a maximum and minimum value. But it does not tell us how to find these extreme values.
Ex1 shows us that the extreme values may occur at an endpoint of the interval or inside the interval. Thus, in order to find the extreme values of a continuous function on a closed interval, we follow the following four-step procedure:

19. Suppose f(x) is cont on [a,b], to find its maximum and minimum values:
Step 1: find f (x)
Step 2: find all c.n. of f in (a,b)
Step 3: Evaluate f at endpoints: a and b,
Evaluate f at all c.n.
Step 4: Compare these values. The largest value will be the max value. The smallest value will be the min value
Note: This method is called the closed
interval method.