Math 1205 Calculus/Sec. 3.1 Extreme Values of Functions
I.
Absolute Extrema
Defn: Let f be a function with domain D. The f has an absolute maximum value on D at a point c if f(x)<
f(c) for all x in D and an absolute minimum value on D at c if f(x)> f(c) for all x in D.
A. Graphical Examples
1. Using the graph above determine the absolute maximum and the absolute minimum for f on the following
intervals.
a. [a,b]
absolute maximum:
absolute minimum:
b.
[c,e]
absolute maximum:
absolute minimum:
c.
(a,d)
absolute maximum:
absolute minimum:
2. Given f(x)=x2, determine the absolute maximum and the absolute minimum for f on the following intervals.
a.
,
absolute maximum:
absolute minimum:
b.
[-1,2]
absolute maximum:
absolute minimum:
c.
[2,5]
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
d. (-3,0)
3. Given g x  
1
2 , determine the absolute maximum and the absolute minimum for g on the following
x
intervals.
a.
,
absolute maximum:
absolute minimum:
b.
[-1,2]
absolute maximum:
absolute minimum:
c.
[1,2]
absolute maximum:
absolute minimum:
II.
Theorems and Definitions
1. Thm 1: If f is continuous at every point of a closed interval I, then f assumes both an absolute
maximum, M and an absolute minimum, m somewhere in I. (m<f(x)<M for all x in I)
2. Defn: Local Extrema (Relative Extrema)
A function f has a local maximum at an interior point c of its domain if f(x)< f(c) for all x in some interval
containing c. A function f has a local minimum at an interior point c of its domain if f(x)> f(c) for all x in
some interval containing c.
3. Defn: An interior point of the domain of a function f where f’ is zero or undefined is a critical point of f.
(A stationary point exists where f’(x)=0 and a singular point exists where f’(x) is undefined)
4. Extrema occur at a critical point or at an endpoint.
III.
1.
2.
3.
4.
5.
6.
IV.
Steps to find absolute extrema of a continuous function on the interval [a,b]
Differentiate f. (find f’)
Determine the critical points. (set f’=0 or undefined)
Determine if the critical point(s) are in the interval [a,b].
Determine f(a), f(b) and f(c) , where c is a critical point.
Compare f(a), f(b) and f(c).
The absolute maximum is the largest function value and the absolute minimum is the smallest function
value.
Examples.
1. Determine the absolute extrema for f(x)=x3-6x2+1 on
a. [-2,3]
Absolute maximum of
at x=
Absolute minimum of
at x=
Absolute maximum of
at x=
Absolute minimum of
at x=
Absolute maximum of
at x=
Absolute minimum of
at x=
b. [-2,4]
c.
(-1,3)
2. Determine the absolute extrema for
gx   x
4
3
1
3
 4x on [-2,2]
Absolute maximum of
at x=
Absolute minimum of
at x=
3.
Determine the absolute extrema for h x  
x  7 . (Domain is implied)
Absolute maximum of
at x=
Absolute minimum of
at x=
Absolute maximum of
at x=
Absolute minimum of
at x=
Absolute maximum of
at x=
Absolute minimum of
at x=
4. Determine the absolute extrema for y=-x3+12x+5 on
a. [-3,3]
b. [-5,5]