0
$\begingroup$

given two functions $f(x)$ and $g(x)$ such that :

1) $f(x) \geq 0$ and $g(x) \geq 0$ for all $x \geq 0$

2) $\lim \limits_{x \to \infty} \dfrac{f(x)}{g(x)} =1$

3) $\dfrac{d}{dx}(\dfrac{f(x)}{g(x)})\Big|_{x=t} \geq 0$ whenever $t \geq x_0$

can we conclude that $f(x) \leq g(x)$ whenever $x \geq x_0$ ?!

  • 0
    Yes, but this site is for professional mathematicians and their PhD students to ask questions about their research. The site to use for questions like these is Mathematics StackExchange; please use that in the future. I will migrate this for you. – 2017-01-31

1 Answers 1

1

Note that 3) implies that $g(x)>0$ if $x\geq x_0$.

First, 3) shows that $f/g$ is non-decreasing in the interval $[x_0;\infty[$ and 2) shows that it tends to 1 as $x$ tends to $\infty$. It is then fairly obvious that $$\dfrac{f(x)}{g(x)}\leq 1$$ for every $x\geq x_0$. Positivity of $g$ then implies that $f(x)\leq g(x)$ if $x\geq x_0$.