AP CALCULUS NOTES
SECTION 5.5 ABSOLUTE MAXIMA AND MINIMA
Relative max. and min. are just high and low points in relation to a local interval. They need not
be the highest and lowest points on the entire graph. Such points are called absolute max and
absolute min.
A.) Definition:

A function f is said to have an absolute maximum on an interval I at x0 if f  x0  is the

largest value of f on I. In other words, f  x0   f  x  for all x in the domain of f on the
interval I.
A function f is said to have an absolute minimum on an interval I at x0 if f  x0  is the

smallest value of f on I. In other words, f  x0   f  x  for all x in the domain of f on the
interval I.
If f has either an absolute maximum or an absolute minimum on I at x0 , then f is said to
have an absolute extremum on I at x0 .
Ex.1.) Use the graph to find x-coordinates of the relative extrema and absolute extrema of f on
0,7 .
EXTREME VALUE THEOREM: If f is a continuous function on a finite closed interval
a, b , then f has both an absolute maximum and an absolute minimum on a, b .

Absolute extremum occur either at the endpoints of the interval or at critical numbers
inside the interval.
Note: If f has an absolute extremum on an open interval  a, b  , then it must occur at a critical
number of f.
Ex.2.) Find the absolute maximum and minimum values of the function
 1 
f  x   x 3  3x 2  1;   , 4  .
The Closed Interval Method:
 2 
1.) Find the critical #’s on  a, b  .
2.) Evaluate f at all critical #’s and
at the endpoints a and b.
3.) The abs. max. occurs at the
largest of these values and the abs.
min occurs at the smallest.
B.) Absolute Extrema on Infinite Intervals: certain conclusions about the existence of absolute
extrema of a continuous function f on  ,   can be drawn from the end behavior.
C.) Absolute Extrema on Open Intervals: examine lim f  x  and lim f  x  .
x a
19.) f  x   x3  3x  2;
 , 
x b
21.) f  x  
x2
;
x 1
 5, 1
THEOREM: Suppose that f is continuous and has exactly one relative extremum at x0 on an
interval I.
1. If f has a relative minimum at x0 , then f  x0  is the absolute minimum of f on I.
2. If f has a relative maximum at x0 , then f  x0  is the absolute maximum of f on I.
Sample Problems:
3.c.)
4.c.)
-8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8
Ex.3.) Find the absolute maximum and minimum values of f  x   10x  2  ln x  on the interval
1, e 2  .