LECTURE 16
21 Critical Points: Properties and Applications
21.1 Absolute Maxima and Minima: Definition
Definition. Suppose the domain of a function fincludes a closed interval ]a, b[. A point cis said to be
•an absolute maximum of fon [a, b] if
f(c)≥f(x) for all xin ]a, b[
•an absolute minimum of fon [a, b] if
f(c)≤f(x) for all xin ]a, b[
Note. Absolute maxima and minima of a function on an open interval, and on an arbitrary set of points, are
defined analogously (e.g., see absolute maxima in S&M.)
Examples
•on the interval [0,4π], the function f(x) = sin(x) has two absolute maxima and two absolute minima (find
them)
•on the interval [1,2], the function
f(x) = 1−xfor 0 ≤x < 1
1 for 1 ≤x≤2
has absolute maxima at the point x= 0 and at all points in the interval [1,2] on the x-axis, but no absolute
minima
21.2 Finite Differences: Definition
Suppose a function fis defined in a neighborhood of a point c.
Exercise. This exercise is an efficient way to clarify the limit formulas for the right-sided and left-sided deriva-
tives.
Let hbe a small, positive number, such that the closed interval [c−h,c +h] is contained in the domain of f.
Write down the slope of the line passing through
•the points (c, f (c)) and (c+h, f(c+h))
•the points (c, f (c)) and (c−h, f(c−h))
Definition. For a small h > 0, the expressions
f(c+h)−f(c)
hand f(c)−f(c−h)
h
are called, respectively, the right-sided and left-sided finite differences (of fat cwith step h). Another term for these
is, a right-sided (left-sided) (f, c, h)-difference.
Note. Finite differences have the following geometric interpretation. A right-sided finite (f , c, h)-difference is the
slope of the line passing through the points
(c, f (c)),(c+h,f (c+h))
Similarly, a left-sided (f, c, h)-difference is the slope of the line passing through the points
(c, f (c)),(c−h,f (c−h))
Note that for both differences, his positive.
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