Quotient of monotone functions is monotone?

Question

Suppose $f,g$ are monotone( say increasing) and differentiable and nonnegative. Both go from $\mathbb{R}^{\geq 1} \to \mathbb{R} $

Is $\frac{f}{g}$ also monotone ?

Did you try using the quotient rule? – Nov 02 '13 at 07:22 — , Nov 02 '13 at 07:22

score 5 · Answer 1 · answered Nov 02 '13 at 08:20

5

$f(x)=2x+\sin x,\ g(x)=x,\ 0\lt x\lt\infty$

answered Nov 02 '13 at 08:20

bof

Newb · Accepted Answer · 2013-11-02T08:04:14.487

0

Counterexample: Let $f(x) = e^x+5, g(x) = x$. $$\frac{f}{g}(x) = \frac{e^x+5}{x}$$

Can you tell me why $\displaystyle{\frac{f}{g}}$ is not necessarily monotone?

edited Nov 02 '13 at 08:04

answered Nov 02 '13 at 07:25

Newb

so you are claiming that $e^xx - e^x < 0 $? in other words, $\frac{1}{x} > 1 $ which is false – Nov 02 '13 at 07:30
I mean it is false your claim. your $f$ is increasing – Nov 02 '13 at 07:31
It's not false at all. Plot the equation on a graph. This should work: http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiJlXngveCIsImNvbG9yIjoiIzAwMDAwMCJ9LHsidHlwZSI6MTAwMH1d – Newb Nov 02 '13 at 07:33
I mean it is false for $x \geq 1 $ – Nov 02 '13 at 07:34
Your question's conditions didn't specify that. Okay, now you've edited them... – Newb Nov 02 '13 at 07:35
I have updated my qs – Nov 02 '13 at 07:35
Then translate the function.... – Nov 02 '13 at 07:38
Why the downvotes? – Newb Nov 02 '13 at 07:59

2 Answers2