Section 4.3
How Derivatives Affect the Shape of a Graph
First derivative is
used for
- Determining where a function is increasing or decreasing.
- Testing whether a critical number is a local maximum or minimum.
Second Derivative is
used for
- Determining concavity of f(x).
- Determine points of inflection (points where the curve changes concavity)
- Determining if a critical number yields a local maximum or minimum.
Increasing/Decreasing
Test
(a) If
on an interval, then f
is increasing on that interval.
(b) If 
on an interval, then f
is decreasing on that interval.
The First
Derivative Test for Local Maximum or Minimum
Suppose that c is a critical number of a continuous function f.
(a) If
changes from positive
to negative at c, then f has a local
maximum at c.
(b) If changes from negative
to positive at c, then f has a local
minimum at c.
(c)
If does not change sign
at c, then f has no local maximum or
minimum at c.
___________________ ___________________ __________________
Concave Up
If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I.
Concave Down
If the graph of f lies below all of its tangents on an interval I, then it is called concave downward on I.
Point Of Inflection (POI)
A point on a curve is called an inflection point (or POI) if f is continuous there and the curve changes concavity at that point.
___________________ ___________________ __________________
Concavity
Test (Using the second derivative.)
(a) If
>0 for all x in I, then the graph of f is concave upward
on I.
(b) If <0 for all x in I, then the graph of f is concave downward
on I.
The Second
Derivative Test for Local Maximum or Minimum.
(a) If
and 
, then f has a local minimum at c.
(b) If and 
, then f has a local maximum at c.