Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 1963 by 123456 last updated on 26/Oct/15

f:[0,+∞)→R,a:N→R a_(n+1) =f(a_n )−a_n f(x)≥f(y),∀x≥y≥0 does a_n ≥a_m ,∀n≥m≥0?

$${f}:\left[\mathrm{0},+\infty\right)\rightarrow\mathbb{R},{a}:\mathbb{N}\rightarrow\mathbb{R} \\ $$$${a}_{{n}+\mathrm{1}} ={f}\left({a}_{{n}} \right)−{a}_{{n}} \\ $$$${f}\left({x}\right)\geqslant{f}\left({y}\right),\forall{x}\geqslant{y}\geqslant\mathrm{0} \\ $$$$\mathrm{does} \\ $$$${a}_{{n}} \geqslant{a}_{{m}} ,\forall{n}\geqslant{m}\geqslant\mathrm{0}? \\ $$

Answered by prakash jain last updated on 26/Oct/15

Counter example a_0 =1 f(1)=0⇒f(a_0 )=0 a_1 =0−1=−1 a_1 <a_0

$$\mathrm{Counter}\:\mathrm{example} \\ $$$${a}_{\mathrm{0}} =\mathrm{1} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0}\Rightarrow{f}\left({a}_{\mathrm{0}} \right)=\mathrm{0} \\ $$$${a}_{\mathrm{1}} =\mathrm{0}−\mathrm{1}=−\mathrm{1} \\ $$$${a}_{\mathrm{1}} <{a}_{\mathrm{0}} \\ $$

Commented by Rasheed Soomro last updated on 27/Oct/15

a_0 =1 Assumed? f(1)=0 How? Is it also assumption? Question for sake of understanding/Knowledge. Not objection.

$${a}_{\mathrm{0}} =\mathrm{1}\:{Assumed}? \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0}\:{How}?\:{Is}\:{it}\:{also}\:{assumption}? \\ $$$${Question}\:{for}\:{sake}\:{of}\:{understanding}/{Knowledge}. \\ $$$${Not}\:{objection}. \\ $$

Commented by prakash jain last updated on 27/Oct/15

Values assumed which satisfy the given conditions in question to create a counter example. We can also use a_(1 ) and a_2 and use differnt values to create a counter example as long as f(a_1 )−a_1 <a_1 f(a_1 )<2a_1

$$\mathrm{Values}\:\mathrm{assumed}\:\mathrm{which}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{given} \\ $$$$\mathrm{conditions}\:\mathrm{in}\:\mathrm{question}\:\mathrm{to}\:\mathrm{create}\:\mathrm{a}\:\mathrm{counter} \\ $$$$\mathrm{example}. \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{also}\:\mathrm{use}\:{a}_{\mathrm{1}\:} \:\mathrm{and}\:{a}_{\mathrm{2}} \:\mathrm{and}\:\mathrm{use}\:\mathrm{differnt} \\ $$$$\mathrm{values}\:\mathrm{to}\:\mathrm{create}\:\mathrm{a}\:\mathrm{counter}\:\mathrm{example}\:\mathrm{as}\:\mathrm{long}\:\mathrm{as} \\ $$$${f}\left({a}_{\mathrm{1}} \right)−{a}_{\mathrm{1}} <{a}_{\mathrm{1}} \\ $$$${f}\left({a}_{\mathrm{1}} \right)<\mathrm{2}{a}_{\mathrm{1}} \\ $$

Commented by Rasheed Soomro last updated on 27/Oct/15

THANK^S !

$$\mathcal{THANK}^{\mathcal{S}} \:! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com