differentiability and extreme points question

2.b)

f is continues in [0,1] and differentiable in (0,1)

f(0)=0 and for x\in(0,1) |f'(x)|<=|f(x)| and 0<a<1

prove:

(i)the set {|f(x)| : 0<=x<=a} has maximum

(ii)for every x\in(0,a] this innequality holds $\displaystyle \frac{f(x)}{x}\leq max{|f(x)|:0<=x<=a}$

(iii)f(x)=0 for $\displaystyle x\in[0,a]$

(iii)f(x)=0 for $\displaystyle x\in[0,1]$

in each of the following subquestion we can use the previosly proves subquestion.

Re: differentiability and extreme points question

Once again, you have posted what looks like a homework with no indication of your own attempt to solve it.

Re: differentiability and extreme points question

my thought on each one:

i was given continuety on closed section and deffentiabilty on open section

so i can use here mean value theory

rolls thery

and weirshtrass

etc...

regarding a :

in order to have maximum by weirshrass its continues on closed section so i have a maximum and minimum between

but the question asks for total maximum(dont know the proper term)

so i dont know what t do next

?

its not home work

it me trying to solve a test from previos semester to which i dont have an answer

so i ask for tip regarding each one to help me solve it

Re: differentiability and extreme points question

Quote:

Originally Posted by

**transgalactic** my thought on each one:

i was given continuety on closed section and deffentiabilty on open section

so i can use here mean value theory

rolls thery

and weirshtrass

etc...

regarding a :

in order to have maximum by weirshrass its continues on closed section so i have a maximum and minimum between

but the question asks for total maximum(dont know the proper term)

so i dont know what t do next

?

its not home work

it me trying to solve a test from previos semester to which i dont have an answer

so i ask for tip regarding each one to help me solve it

It is not our job to provide solutions to tests, exams etc. from a previous semester. It is the job of your institute to provide the solutions. If they are unwilling to provide the solutions, there is probably a very good reason for that .....

Re: differentiability and extreme points question

i dont want you to write a full solution

only guidance.i have written above what i know about general ways

could you say in general what to do from here?