Mathematics Asked on January 7, 2022

If $f: [0,1] to mathbb{R}$ is differentiable, $g$ is continuous, and

$f'(t) = g(f(t)),$ then $f$ is monotonic.

Example 1: Suppose $g(x) = x Rightarrow f’ = f.$ Then $f = ce^x$ which is indeed monotonic.

Example 2: Suppose $g(x) = x^n Rightarrow f’ = f^n$ for $n ne 1.$ Then $df/f^n = dx Rightarrow x+C = f^{1-n}/(1-n) Rightarrow f = ((1-n)x + C)^{1/(1-n)},$ monotonic again.

It seems bizarre that $g$ can be anything we want, and yet the result will be true. Since $f$ is continuous, it suffices to prove $f$ is injective. So suppose $f(a) = f(b)$ with $a<b.$

Avenue 1: Then $f'(a) = g(f(a)) = g(f(b)) = f'(b).$ But what’s next? This seems like a dead end.

Avenue 2: There exists $c in (a,b)$ such that $f'(c) = 0$ by Rolle’s Theorem. Thus, $g(f(c)) = 0.$ But what’s next? Looks like another dead end again.

Is there a 3rd avenue? I’ve been looking for one. Until I find it, I cannot believe the result.

Update: The answer here solves my question. The question also appeared previously on MSE, but I do not like the method used there since it seems to assume $f’$ is continuous.

This is an explanation of an answer found here, which works but leaves out a few details.

Suppose $f(a) = f(b), a<b.$ It will suffice to show $f$ is constant on $[a,b].$ Let $gamma$ be the path parameterized by $gamma(t) = f(t), t in [a,b].$ Then $$int_a^b f'(t)^2 , dt = int_a^b g(f(t))f'(t) = int_{gamma} g(z) , dz = 0$$ since $g$ is continuous and $gamma$ is a closed loop due to the fact that $gamma(a) = f(a) = f(b) = gamma(b).$

Thus, $f'(t) = 0$ on $[a,b],$ which implies $f$ is constant on $[a,b].$

Answered by Display name on January 7, 2022

2 Asked on February 24, 2021

combinatorial game theory combinatorics discrete mathematics game theory

3 Asked on February 24, 2021 by clarinetist

1 Asked on February 24, 2021 by probafds123

1 Asked on February 23, 2021 by lockedscope

2 Asked on February 23, 2021 by user12588

1 Asked on February 23, 2021 by mike-pinnsoneault

0 Asked on February 23, 2021 by opensauce

1 Asked on February 23, 2021 by antonio-maria-di-mauro

3 Asked on February 23, 2021

2 Asked on February 23, 2021

2 Asked on February 23, 2021 by user723228

1 Asked on February 23, 2021

measure theory probability probability distributions probability theory random variables

2 Asked on February 22, 2021 by taylor-rendon

1 Asked on February 22, 2021

3 Asked on February 22, 2021 by believer

2 Asked on February 22, 2021 by inquirer

Get help from others!

Recent Questions

- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?
- Does Google Analytics track 404 page responses as valid page views?

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?

© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP