Name______
Math 160 – Section 3.1
Increasing and Decreasing Functions
Critical Numbers (points)
Read Section 3.1. In the blue boxes of your book, you have the information needed to complete the following statements.
Let f(x) be a continuous function on the interval I
Let a and b, be two numbers in the interval I
1. f is Increasing over I
If for a < b thenf(a) ...... f(b) Show graph here
2. f is decreasing over I
If for a < b then f(a) ...... f(b)Show graph here
3. Theorem I - Derivative Criteria for Increasing and Decreasing functions
If f ’ > 0 for all x in I, then f is ...... over I
Why is this true?
Think of tangent lines.
If f ‘ < 0 for all x in I, then f is ...... over IWhy is this true?
Think of tangent lines.
4. Critical numbers
x = c is a critical number if f ‘(c) ...... or f ‘ (c) ......
5. Critical points
(c, f(c)) is a critical point if f ‘(c) ...... or f ‘ (c) ......
Show a graph of a function with a Horizontal Tangent Line (HTL) at x = c,
and one with a Vertical Tangent Line (VTL)at x = c.
6. Summarize definition of a relative minimum.Show a graph.
7. Summarize definition of a relative maximum.Show a graph.
8. Answer True or False for each of the following statements:
(1) If (c,f(c)) is critical point then it is either a maximum or a minimum point
Justify with a graph.
(2) If (c,f(c)) is a maximum or a minimum point then (c,f(c)) is a critical point
Justify with a graph.
9. Theorem 3: First derivative test
Let P(c, f(c)) be a critical point, then
(1) P is a relative maximum if
f ‘(c) ...... 0 to the ...... of cand
f ‘(c) ...... 0 to the ...... of c
That is, the sign of the first derivative changes from ..... to ..... around c.
Show graphs here.
(2) P is a relative minimum if
f ‘(c) ...... 0 to the ...... of cand
f ‘(c) ...... 0 to the ...... of c
That is, the sign of the first derivative changes from ..... to ..... around c.
Show graphs here.
(3) P is not a relative extremum if
The sign of the first derivative ...... around c.
Show graphs here.