Additional Topics from 4.2 and 4.3
4.2 A function, f(x) has a local minimum at c if there is some interval, (a, b) containing c
for which f(c) < f(x) on (a, b). This is also called a relative minimum. A local or relative
maximum is defined similarly.
Fermat's Theorem: If a continuous function, f(x), has either a local minimum or a local
maximum at c then either f '(c)=0 or f '(c) does not exist.
These are the type of extreme values we find with sign charts.
f(x) has an absolute minimum for the interval [a, b] at c in [a, b] if
f(c) < f(x) for all x in [a, b]. Similarly, f has an absolute maximum at c if f(c)>f(x) for all
x in [a, b].
A point can be both a local and absolute extreme point or it can be one and not the other.
Theorem: If the function, f(x), is continuous on the closed bounded interval, [a, b], then it
has both an absolute minimum and an absolute maximum on [a, b].
An absolute extreme point of a continuous function must occur at either a local extreme
point or an endpoint of the closed bounded interval.
The method for finding the absolute extreme points without a graph is to compare the
function values at all critical numbers in the interval and at the endpoints.
Example: Find the absolute max and absolute min of f(x) on the given interval, [a, b].
3
f ( x)  x  3x
f '( x)  3x
2
2
 9x  4
 6 x  9  3 ( x  3 )( x  1)
f has critical numbers x=-1 and x=3
a) [a, b] = [0, 5] Make a table of the critical values in the given interval and the endpoints
0 and 5.
x f(x)
0
f(0)=4
3
f(3)=-23
5
f(5)=9
So the absolute minimum is -23 at x=3 and the absolute maximum is 9 at x=5.
b) [a, b] = [-3, 4]
x f(x)
-3 f(-3)=-23
-1 f(-1)=9
3 f(3)=-23
4 f(4)=-16
So the absolute minimum is -23 occurring at both x= -3 and x=3. The absolute maximum
is 9 at x = -1.