Question

Please show all steps and explain every line of proof.

show that if f:[a,b] -> R is differentiable on a closed interval [a,b] and if f' is continuous on [a,b], then f is lipshitz on [a.b]

Answer #1

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
definition of f being uniformly continuous on its domain? (This
definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
is uniformly...

Let f: R --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.

Show all the steps and explain. Don't skip steps and please
clear hand written
f(x)=x^m sin(1/x^n) if
x is not equal 0 and f(x)=0 if x =0
(a) prove that when
m>1+n, then the derivative of f is continuous at 0
limit x to 0 x^n
sin(1/x^n) does not exist? but why??? please explain it should be
0*sin(1/x^n)

Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous
such that f is twice differentiable on (a, b), meaning f is
differentiable on (a, b), and f' is also differentiable on (a, b).
Suppose further that there exists c ∈ (a, b) such that f(a) >
f(c) and f(c) < f(b).
prove that there exists x ∈ (a, b) such that f'(x)=0.
then prove there exists z ∈ (a, b) such...

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

show proof with all explanations please!! will give a like.
theorem: Suppose R is an equivalence of a non-empty set A. Let
a, b be within A.
Then [a] does not equal [b] implies that [a] intersect [b] =
empty set

proof of 'f : R->R is concave if f''(x) <= 0 for
all x'

Let F be continuous, show that f([a,b]) is a closed
interval.

Numerical analysis problem, please show all steps thank you.
A four times continuously differentiable function f is given by
the following data: f(1.1)=2, f(1.3)=1.5, f(1.5)=1.2, f(1.7)=1.6.
Assume that |f ''''(t)|=<100 for 1.1<t<1.7. Find the
estimate for f '' (1.3). Give an error bound.

Please show the proof that:
Either [a]=[b]
or [a] *union* [b] = empty set
this will be proof by contrapositibe but please show work:
theorem: suppose R is an equivalence of a non-empty set A. let
a,b be within A
then [a] does not equal [b] implies that [a] *intersection*
[b] = empty set

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