The Real And Complex Number Systems

Derivatives
Real-valued functions
In each following exercise assume, where mecessary, a knowledge of the formulas for
differentiating the elementary trigonometric, exponential, and logarithmic functions.
5.1 Assume that f is said to satisfy a Lipschitz condition of order at c if there
exists a positive number M (which may depend on c) and 1 ball Bc such that
|fx fc| M|x c|
whenever x Bc, x c.
(a) Show that a function which satisfies a Lipschitz condition of order is continuous
at c if 0, and has a derivative at c if 1.
Proof: 1. As 0, given 0, there is a /M 1/ such that as
x c , c Bc, wehave
|fx fc| M|x c| M .
So, we know that f is continuous at c.
2. As 1, consider x Bc, andx c, wehave
fx fc
x c M|x c| 1 0asx c.
So, we know that f has a derivative at c with f c 0.
Remark: It should be note that (a) also holds if we consider the higher dimension.
(b) Given an example of a function satisfying a Lipschitz condition of order 1 at c for
which f c does not exist.
Solution: Consider
||x| |c|| |x c|,
we know that |x| is a function satisfying a Lipschitz condition of order 1 at 0 for which
f 0 does not exist.
5.2 In each of the following cases, determine the intervals in which the function f is
increasing or decreasing and find the maxima and minima (if any) in the set where each f is
defined.
(a) fx x 3 ax b, x R.
Solution: Since f x 3x 2 a on R, we consider two cases: (i) a 0, and (ii) a 0.
(i) As a 0, we know that f is increasig on R by f 0onR. In addition, if f has a
local extremum at some point c, then f c 0. It implies that a 0andc 0. That is,
fx x 3 b has a local extremum at 0. It is impossible since x 3 does not. So, we know
that f has no maximum and minimum.
(ii) As a 0, since f 3x 2 a 3 x a/3 x a/3 , we know that
f x :
, a/3 a/3 , a/3 a/3 ,
0 0 0
which implies that
fx :
, a/3 a/3 , a/3 a/3 ,
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