Warning about a difference in terminology
I (Dave Lantz) prefer a definition of "local"
maximum (or minimum, but I will phrase this statement
in terms of maximum) that is different from the one in
Stewart's text. The text says:
- An x-value
c is a local maximum of the function f
if there is an interval (c-d,c+d) (for some
d>0) around
c such that, for each x in the interval,
f(x) is defined and f(x) is less than
or equal to f(c).
This implies two things:
- The statement of Fermat's theorem given in the
text is very simple (and correct); and
- An endpoint of (an interval in) the
domain of f may be an absolute maximum of
f but cannot be a local maximum.
But I would prefer that every absolute maximum also
be a local maximum, even if it is an (included) endpoint
of the domain, and I am willing to pay the price
of needing a more complicated statement of Fermat's
theorem. Thus, my definition of local maximum is as
follows:
- An x-value
c is a local maximum of the function f
if there is an interval (c-d,c+d) around
c such that, for each x in the interval,
if f(x) is defined, then
f(x) is less than or equal to f(c).
As a result, the correct statement of Fermat's
theorem must be:
- If f has a local maximum or minimum at c,
if f(x) is defined at each x-value in
an interval around c,
and if f'(c) exists, then f'(c)=0.
Moreover, the text is not clear, when using the term
"maximum", whether it refers to the x-value
c (an element of the domain of f),
which is more common in calculus classes;
or to the corresponding y-value f(c) (an
element of the
range of f), which probably makes more sense.
I will try to use the former meaning of the term
consistently, but I may slip.
My students should know that I do not consider either of
these terminological matters important, and that they will
not affect grading on exams (and should not on homework,
either, but the grader may slip). There are many more
important, difficult and confusing matters, even of
terminology, in this course;
the ones described here are unlikely to cause confusion.
Questions? Send me e-mail:
dlantz@mail.colgate.edu
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