1. prove or desprove claims

determine if the following are correct or wrong
if correct prove if not then show counter example:
a)if $lim_{x->\infty}f(x)=\infty$ and $g(x)>0$ for $x\in R$ then
$lim_{x->\infty}f(x)g(x)=\infty$
couldnt find counter example ,i thought f(x)=x but there is not g(x)
to desprove it

b) $2xArctanx>ln(1+x^{2})$ for x>0
i was told told to use the theore, of lagrange
in which the derivative in some point is the slope between two points
$f'(c)=\frac{f(b)-f(a)}{b-a}$ but i dont know how to use it

c)f(x) if bound from the top on (0,1{]}
if f(x) is continues from the left of x=1 then sup f((0,1))>=f(1)

d)if f(x) is integrabile on {[}a,b{]} and if $\intop_{a}^{b}f(x)dx>1$
then there is $c\in(a,b)$
so $\intop_{a}^{b}f(x)dx=1$

2. Re: prove or desprove claims

Originally Posted by transgalactic
[LEFT]
determine if the following are correct or wrong
if correct prove if not then show counter example:
a)if $lim_{x->\infty}f(x)=\infty$ and $g(x)>0$ for $x\in R$ then
$lim_{x->\infty}f(x)g(x)=\infty$
couldnt find counter example ,i thought f(x)=x but there is not g(x)
to desprove it
(a) how about $f(x) = x$ and $g(x) = \frac{1}{x^2+1}$ for a counter-example?

3. Re: prove or desprove claims

determine if the following are correct or wrong
if correct prove if not then show counter example:
a)if $lim_{x->\infty}f(x)=\infty$ and $g(x)>0$ for $x\in R$ then
$lim_{x->\infty}f(x)g(x)=\infty$
couldnt find counter example ,i thought f(x)=x but there is not g(x)
to desprove it
b) $2xArctanx>ln(1+x^{2})$ for x>0
i was told told to use the theore, of lagrange
in which the derivative in some point is the slope between two points
$f'(c)=\frac{f(b)-f(a)}{b-a}$ but i dont know how to use it
c)f(x) if bound from the top on (0,1{]}
if f(x) is continues from the left of x=1 then sup f((0,1))>=f(1)
d)if f(x) is integrabile on {[}a,b{]} and if $\intop_{a}^{b}f(x)dx>1$
then there is $c\in(a,b)$
so $\intop_{a}^{b}f(x)dx=1$

(a)
I like your idea. How about $f(x)=x, g(x)=1/x^2$ ?

(d)
is the following function continuous? $g(c) = \int_a^c f(x) dx$

if so you can use the intermediate value theorum, as its trivial to show that there are values of g(c) on either side of 1.

4. Re: prove or desprove claims

yes thanks
1 down 3 to go

5. Re: prove or desprove claims

regarding b) i was told to turn $2xArctanx>ln(1+x^{2})$ into $2xArctanx-ln(1+x^{2})>0$
$f(x)=2xArctanx-ln(1+x^{2})$
so i need to prove that f(x)>0 for x>0
f(0)=0
now i need to show that f'>0 so f will be monotonicly increasing
$f'(x)=2Arctanx+2\frac{1}{1+x^2}- \frac{2x}{1+x^{2}}$
how to conclude that f' is positive for x>0