Monotonicity and Extrema of Functions
See or remember these geometric
and algebraic previews of
calculus (Chapters 2 to 4 in Volume 3) for an easy informal examples of
- the monotonicity analysis of functions y = f(x), that is where they are
increasing or decreasing, with the where referring to intervals,
- the location of extrema: local, end-point and absolute maxima and minima,
and
- the sign analysis not of functions, but of formulas for their slopes.
Then read the following formal definitions for skill & concept perfection
if not development.
If you are Quebcee math 436, the ability to solve problems
is with regret more for passing the final examination than theory.
Absolute Extrema
In the following, suppose y = f(x) is real-valued function of a real variable
x.
- Definition: A point x = a in the domain of f is an absolute
maximum of f if and only if for all other points b in the domain of
f, f(a) > f(b).
- Definition: A point x = a in the domain of f is an absolute
minimum of f if and only if for all other points b in the domain of
f, f(a) < f(b).
- Definition: A points x = a in the domain of f is an absolute
extremum if and if only x = a is an absolute maximum or an absolute
minimum of f.
Food for thought: Strict absolute maxima and minima might be defined
by insisting on strict inequalities (inequalities without the option of
equality) above. However, I have not seen that done elsewhere.
Local Extrema
In the following, suppose y = f(x) is real-valued function of a real variable
x. We will say here that an interval is proper if and only if its length is
nonzero, that is, if and only if, the interval does not consist of a single
point.
- Definition: A point x = a in the domain of f is an local maximum
of f if and only if there is a proper interval I centered at x = a such for
all other points b in the interval I and the domain of f, f(a) >
f(b).
- Definition: A point x = a in the domain of f is an local minimum
of f if and only if for all other points b in the interval and in the domain
of f, f(a) < f(b).
- Definition: A points x = a in the domain of f is an local
extremum if and if only x = a is a local maximum or an local minimum of
f.
Right and Left End-Point Extrema
More Maxima and Minima.
In the following, suppose y = f(x) is real-valued function of a real variable
x whose domain is a finite or semi-infinite interval (or a union of such
intervals). Suppose a in domain f is an end-point of the interval of
definition of f (or one of the intervals of definition). Then
- Definition: The point x = a is an end-point maximum if and
only if x = a is an absolute or local maximum of y = f(x).
- Definition: The point x = a is an end-point minima if and
only if x = a is an absolute or local miniimum of y = f(x).
Since intervals have left and right end-points, we can also speak of
and formally define left and right end-point extrema. There we may be
trespassing further into too much jargon or terminology
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