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CURVE SKETCHING

is undefined at

is decreasing when

CRITICAL/EXTREME POINT:

Another important concept needed in curve sketching is that of a critical point. If
is in the domain of and either or is not defined, then is called a
critical value of the function, and is called a critical point. A critical point
may be a maximum point, minimum point, or neither.

A relative (or local) maximum point is a critical point where the function changes from
increasing to decreasing.

A relative (or local) minimum point is a critical point where the function changes from
decreasing to increasing.

The critical point is neither a maximum nor a minimum if the function does not change
from increasing to decreasing (or vice versa) at the critical point.


is defined and so is a critical point. • is increasing before and decreasing after so is a	• is defined and is undefined so (c,f(c)) is a critical point. • is decreasing before and increasing after so is a relative minimum	• is defined and so (0,0) is a critical point. • is increasing before and increasing after so is neither a max nor a min.

To locate the critical points on the graph:

1. Take the first derivative of the function and determine the values where
or is undefined.
2. If c is in the domain of, then is a critical point.

For example:

Earlier we found that , when and , and is defined
everywhere. Since is defined for both and , we have found two critical
values.

Substituting these values into our original function, we find that

Thus, we have found critical points at (1,-6) and (-1,6).

You can determine whether these points are local maximum points, local minimum
points, or neither, using either the first derivative test or the second derivative test (the
second derivative test will be explained in the next section).

FIRST DERIVATIVE TEST

1. Determine where the function is increasing or decreasing.
2. If the function is increasing before the critical value and decreasing after the critical
value, then the critical point is a local maximum. If the function is decreasing before
the critical value and increasing after, the point is a local minimum. Otherwise, the
critical point is neither a maximum nor a minimum.

For example: Earlier we found critical points for at (1,-6) and (-1,6). To
determine whether these points are local maximums or minimums, use the first
derivative test.

First, determine where is increasing and decreasing.
The critical values found above were and .

From our previous example we found thatis increasing on (-∞,1), decreasing on
(-1,1), and increasing on (1,∞). Thus, we had the following number line:

At the critical value , the function changes from increasing to decreasing.
Therefore, (-1,6) is a local maximum.

At the critical value , the function changes from decreasing to increasing.
Therefore, (1,-6) is a local minimum.

CONCAVITY:

Concavity describes the general "cupping" of a function at a particular point or interval.
When the slope of the tangent to the curve is increasing over an interval ( is
increasing - i.e. ), the function is concave up. When the slope of the tangent to
the curve is decreasing over an interval ( is decreasing, i.e. ), the function
is concave down.

		Slope of tangent lines is decreasing (() on the interval

Slope of tangent lines is increasing () on the interval

Notice that the point A is a critical point (since the slope of the tangent line is 0 at A) and
is concave up at A. We can see that A is a local minimum. Also, the point B is a
critical point (since the slope of the tangent line is 0 at B) andis concave down at B.
We can see that B is a local maximum.

Concavity can help us determine if a critical point is a local maximum or a minimum.
The following is the second method for determining whether a critical point is a local
maximum or a minimum.

SECOND DERIVATIVE TEST

If is a critical point, then:

1. If , the function is concave down at that point and thus is a local
maximum point.
2. If , the function is concave up at that point and thus is a local
minimum point.
3 If , then the second derivative test fails to determine if the point is a local maximum
or a minimum. In this case, the first derivative test mentioned earlier should be used.

Example: Let's go back to the function

Our critical points were (-1,6) and (1,-6).
Using the second derivative test we obtain the following:

Since , the function is concave down at (-1,6), thus (-1,6)
is a local maximum.

Since , the function is concave up at (1,-6), thus (1,-6) is
a local minimum.

We see that we get the same results using the Second Derivative Test as we do using
the First Derivative Test.

INFLECTION POINTS:

An inflection point is a point on a graph (in the domain of) where concavity
changes from concave up to concave down, or vice-versa. Concavity can change at
values where or is undefined.

To find inflection points:

1. Determine where the function is concave up and where it is concave down:
a) Determine the value(s) of x where or is undefined.
b) Order the values found above in increasing order and plot them on a number line.
c) For every interval between two consecutive values, choose a test value in that
interval.
d) Determine the value ofat the test value.
e) If at the test value, then is concave up on that interval. If at
the test point, thenis concave down on that interval.

2. If the function changes from concave up to concave down (or vice-versa) at
and is defined, then is an inflection point.

Example: Earlier we found that the second derivative of was
. Thus when , and is defined everywhere.

Plotting this on a number line we get:

For the interval (-∞,0), choose to be our test point:
Since , , is concave down on the interval (-∞,0).

For the interval (0, ∞), choose to be our test point:
Since , is concave up on the interval (0, ∞).

Labeling our number line we get:

Since changes from concave down to concave up at and is defined at ,
,the point is an inflection point.

Remember that values where or is undefined are only potential places
where the graph can change concavity. It is possible, however, that the function may
not change concavity at those values.

For example, consider . Then and , hence
when ..

Plotting this on a number line we get:

For the interval (-∞,0), choose to be our test point:
Since , , is concave up on the interval (-∞,0).

For the interval (0,∞), choose to be our test point:
Since , is concave up on the interval (0, ∞).

Labeling our number line we get:

In this example is concave up when and concave up when .
Concavity did not change at , so the point (0,0) is not an inflection point.