minimum there. Note that at
f has neither a maximum nor a minimum, since the derivative
does not change signs from the left to the right of the point.
Absolute Extrema
If
is the largest value that f attains on some interval
containing c, then M is called
the global maximum of f on
. Similarly, if
is the smallest value that f attains on
some interval
containing c, then M is called the global minimum of f on
.
There is no reason to expect that an arbitrary function has a global maximum or minimum value
on a given interval. However, the Extreme Value Theorem guarantees that a function does have a
global maximum and a global minimum on any closed interval on which it is continuous. On
such an interval, both of the global extrema must occur at either a critical point or at an endpoint
of the interval.
The candidate test gives a procedure for finding these global extrema on a closed interval
:
1. Check that f is continuous on
.
2. Find the critical numbers of f between a and b.
3. Check the value of f at each critical number, at a and at b.
4. The largest value found in the previous step is the global maximum, and the smallest
value found is the global minimum.
Concavity and Inflection Points
The graph of a function f is concave up when its derivative
is increasing, and it is concave
down when
is decreasing. Since the relationship of
to
is the same as the relationship
of
to f, we can determine on which intervals
is increasing (or decreasing) by checking
where
is positive (or negative). Therefore, the criteria for f being concave up or down can be
restated in terms of
: f is concave up when
is positive, and concave down when
is
negative.
A point at which a function changes concavity (from up to down or down to up) is called a point
of inflection. These can be found in a completely analogous manner to how local extrema are
located using the first derivative test: find where the second derivative is 0 or undefined, and test
points on either side to determine if concavity is changing.
Second Derivative Test
In addition to providing information about concavity and inflection points, the second derivative
of a function can also help determine whether a critical point represents a relative maximum or
minimum. Specifically, suppose f has a critical point at
.
Then: