AP Calculus AB
Section 1.6 Notes
I.
Def:
Let I denote the interval a, b . A functions f is increasing on I if f x1   f x2  whenever a  x1  x2  b.
Let I denote the interval a, b . A functions f is decreasing on I if f x1   f x2  whenever a  x1  x2  b.
(If f x1   f x2  for x1  x2 , it is nondecreasing. If f x1   f x2  for x1  x2 , it is nonincreasing.)
Ex:
Def:
Let f be a function and x 0 a point of its domain.
A.
x0 is a stationary point of f if f x   0.
Ex: (graph above)
B.
C.
D.
E.
x 0 is a local maximum point of f if f x0   f x  for all x in some open interval containing x.
The number f x0  is a local maximum value of f.
Ex: (graph above)
x 0 is a local minimum point of f if f x0   f x  for all x in some open interval containing x.
The number f x0  is a local minimum value of f.
Ex: (graph above)
x 0 is a global maximum point of f if f x0   f x  for all x in the domain of f.
The number f x0  is a global maximum value of f.
Ex: (graph above)
x 0 is a global minimum point of f if f x0   f x  for all x in the domain of f.
The number f x0  is a global minimum value of f.
Ex: (graph above)
F.
II.
x 0 is a critical point of f if either
(1)
f x0  is undefined because there is a corner in the graph at x 0 .
Ex:
(2)
f x0  is a vertical tangent line at x 0 .
Ex:
(3)
f x0   0 , and thus is a stationary point.
Ex:
Facts about f  .
A.
If f a   0 , then f is increasing at x  a . If f a   0 , then f is decreasing at x  a .
Ex:
B.
C.
If f increases at x  a , then f a   0 . If f decreases at x  a , then f a   0 .
Ex:
Suppose f  exists for all x in the domain of f. Every local max or local min point x 0 is a root of f  .
 f x0   0.
Refer to graphs on front for example.
D.
FIRST DERIVATIVE TEST:
Suppose f x0   0 . (Every root of f x  is a stationary point, so possibly is a local max or min point.)
(1)
If f x   0 for all x  x0 and f x   0 for all x  x0 , then x 0 is a local minimum point.
(2)
If f x   0 for all x  x0 and f x   0 for all x  x0 , then x 0 is a local maximum point.
Ex:
-
IV.
Def:
A.
B.
C.
The graph of f is concave up at x  a if the slope function f  is increasing at x  a .
The graph of f is concave down at x  a if the slope function f  is decreasing at x  a .
An input value at which a graph’s concavity changes is an inflection point of the graph.
Ex:
If f a  0 , then f  is increasing at x  a .
If f a  0 , then f  is decreasing at x  a .
The roots of f  are the stationary points of f  .
If f a  0 , then f is concave up at x  a .
If f a  0 , then f is concave down at x  a .
The roots of f  are the inflection points of f.