# Thread: derivative inequality

1. ## derivative inequality

can someone help me??!

I have this problem..

if $x(t)^{'}$ is a nonnegative and decreasing function and

$x(t)$ is a positive and nondecreasing

so..

$\frac{x^{'}(t)}{x(t)}\leqslant \frac{1}{t}$ for all $t\geqslant b$ ??!

I tried to prove this problem but at this step

$x(t)-x(b)\geqslant(t-b)x(t)^{'}$

I don't know how to get the rest of proof from this step..

can someone help me to get the rest of proof??

thanks for any help

2. What is $x(t)$, what is $x'(t)$, on what interval are they defined? What is $b$?

3. ## moredetial

if $x^{'}(t)$ is a nonnegative and decreasing function
and $x(t)$ is a positive and nondecreasing on [b,t]

then

$x(t)-x(b)=\int^{t}_{b}{x^{'}(s)ds}$

since $x^{'}(t)$ is decreasing

$x^{'}(t)\leqslant x^{'}(s)$

this implies

$x(t)-x(b)\geqslant x^{'}(t) \int^{t}_{b}{ds}$

so

$x(t)-x(b)\geqslant(t-b)x^{'}(t)$

i.e.

$\frac{x^{'}(t)}{x(t)}\leqslant \frac{1}{t-b}$.....(*)

now,I just want to get

$\frac{x^{'}(t)}{x(t)}\leqslant \frac{1}{t}$??!

I hope that's clear