Page 111 - Calculus Workbook For Dummies

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Chapter 7: Analyzing Those Shapely Curves with the Derivative

The Second Derivative Test and Local Extrema

With the second derivative test, you use — can you guess? — the second derivative to
test for local extrema. The second derivative test is based on the absolutely brilliant
idea that the crest of a hill has a hump shape (+) and the bottom of a valley has a
trough shape (,).
After you find a function’s critical numbers, you’ve got to decide whether to use the
first or the second derivative test to find the extrema. For some functions, the second
derivative test is the easier of the two because 1) The second derivative is usually
easy to get, 2) You can often plug the critical numbers into the second derivative and
do a quick computation, and 3) you will often get non-zero results and thus get your
answers without having to do a sign graph and test regions. On the other hand, testing
regions on a sign graph (the first derivative test) is also fairly quick and easy, and if the
second derivative test fails (see the warning), you’ll have to do that anyway. As you do
practice problems, you’ll get a feel for when to use each test.

When the second derivative equals zero, the second derivative test fails and you learn
nothing about local extrema. You’ve then got to use the first derivative test to deter-
mine whether there’s a local extremum (the singular of extrema) there.

Q. Take the function from the example in the Tip: If you only have one critical point between
5
3
previous section, g x = 15 x - x , but this a local min and a local max (and no discon-
^ h
time find its local extrema using the second tinuities), it has to be an inflection point,
derivative test. and if you have a single critical number
between two known maxes (see problem
A. The local min is at x = –3 and the local 7), the only possibility for the middle criti-
max is at x = 3. cal number is a local min (and vice-versa).
So in these cases, it really doesn’t matter
You’ve probably figured out that to use if the second derivative test fails with the
the second derivative test you need the middle critical number. If this not-by-the-
second derivative: book reasoning doesn’t work for your calc
teacher, you might say (with just a touch
5
3
g x = 15 x - x
^ h
of sarcasm in your voice), “Oh, so in other
2
g x = 45 x - 5 x 4 words, you’ve got something against logic
l ^ h
x =
gm ^ h 90 x - 20 x 3 and common sense.”
Now all you do is plug in the critical
numbers of g from the example in the
last section:
3 =
gm ^ - h 270
0 =
gm ^ h 0
3 = -
gm ^ h 270
The fact that gm ^ - 3h is positive tells you
that g is concave up (,) there, and thus
that there’s a local min. And the fact that
gm ^ 3h is negative tells you that g is con-
cave down (+) at x = 3, and, therefore,
that there’s a local max there. And, while
0 =
it may seem that gm ^ h 0 confirms what
you figured out previously (that there’s
neither a min nor a max there), you actu-
ally learn nothing when the second deriv-
ative is zero; you have to use the first
derivative test.