Is there any continuous function that satisfies the following conditions?

Question

Is there any continuous function $y=f(x)$ on $[0,1]$ that satisfies the following conditions? $$ f(0) = f(1) = 0, $$ and $$ f(a)^2-f(2a)f(b)+f(b)^2<0, $$ for some $0<a<b<1$.

I tried to test with several functions (with different $a,b$) but non of them satisfied.

Any help is appreciated. Thank you in advance.

is this for any $0<a<b<1$ or for all of them? – gt6989b Jan 17 '17 at 23:14 — gt6989b, Jan 17 '17 at 23:14
What is the domain of $f$? – Michael Hoppe Jan 17 '17 at 23:22 — Michael Hoppe, Jan 17 '17 at 23:22
For any $0<a<b<1$. Thanks. – Ana Uspekova Jan 17 '17 at 23:25 — Ana Uspekova, Jan 17 '17 at 23:25

score 16 · Accepted Answer · answered Jan 17 '17 at 23:31

16

What you want is a function that is highly peaked around $x=2a$. And you need to enforce the conditions that $f(0)=f(1)=0$; for that you can put in a factor of $x(1-x)$.

So try

$$ f(x) = x(1-x)e^{-20(x-\frac12)^2} $$ with $a=\frac14$ and $b=\frac34$.

There, $$f(a)^2 - f(2a)f(b) + f(b)^2 \approx -0.00766 <0$$

answered Jan 17 '17 at 23:31

Mark Fischler

41,743

Thank you very much for your help! – Ana Uspekova Jan 17 '17 at 23:34

bof · Answer 2 · 2017-01-18T10:47:10.653

16

All you have to do is choose numbers $a,b$ so that $0\lt a\lt 2a\lt b\lt1$ (or $0\lt a\lt b\lt2a\lt1$), then find numbers $y_1,y_2,y_3$ satisfying $y_1^2-y_2y_3+y_3^2\lt0,$ and then construct a continuous function (e.g. a fourth degree polynomial) $f(x)$ with $f(0)=f(1)=0,\ f(a)=y_1,\ f(2a)=y_2,\ f(b)=y_3.$ For example:

$$f(x)=x(x-1)(4x-1)(8x-7),\ a=\frac14,\ b=\frac34$$

edited Jan 18 '17 at 10:47

answered Jan 18 '17 at 00:01

bof

78,265

Nice example! Thank you very much! – Ana Uspekova Jan 18 '17 at 00:58
2

@AnaUspekova: Or you can just draw straight lines to connect the dots required. – user21820 Jan 18 '17 at 05:55
Is there a general approach to solving a problem of this kind? i.e. Find a function f that obeys the following constraints. – curious_cat Jan 18 '17 at 06:25
....or even more specifically: Find a polynomial function that satisfies the following constraints. – curious_cat Jan 18 '17 at 06:25
@curious_cat Are you asking about the Lagrange interpolation formula or something else? – bof Jan 18 '17 at 06:53
@bof Thanks. No, I meant is there a more algorithmic way to come up with such functions given constraints. – curious_cat Jan 18 '17 at 07:29
@curious_cat I have no idea what kind of functions you want to come up with or what kind of constraints you want them to satisfy. Maybe if you could give an example or two I would at least understand the question, though I don't expect to know the answer. – bof Jan 18 '17 at 07:51

Is there any continuous function that satisfies the following conditions?

2 Answers2