# prove or desprove claims

• Aug 28th 2011, 01:08 PM
transgalactic
prove or desprove claims
determine if the following are correct or wrong
if correct prove if not then show counter example:
a)if $\displaystyle lim_{x->\infty}f(x)=\infty$ and $\displaystyle g(x)>0$ for $\displaystyle x\in R$ then
$\displaystyle lim_{x->\infty}f(x)g(x)=\infty$
couldnt find counter example ,i thought f(x)=x but there is not g(x)
to desprove it

b)$\displaystyle 2xArctanx>ln(1+x^{2})$ 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
$\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$ 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 $\displaystyle \intop_{a}^{b}f(x)dx>1$
then there is $\displaystyle c\in(a,b)$
so $\displaystyle \intop_{a}^{b}f(x)dx=1$
• Aug 28th 2011, 01:26 PM
skeeter
Re: prove or desprove claims
Quote:

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

(a) how about $\displaystyle f(x) = x$ and $\displaystyle g(x) = \frac{1}{x^2+1}$ for a counter-example?
• Aug 28th 2011, 01:29 PM
SpringFan25
Re: prove or desprove claims
Quote:

determine if the following are correct or wrong
if correct prove if not then show counter example:
a)if http://latex.codecogs.com/png.latex?...8x%29=%5Cinfty and http://latex.codecogs.com/png.latex?g%28x%29%3E0 for http://latex.codecogs.com/png.latex?x%5Cin%20R then
http://latex.codecogs.com/png.latex?...8x%29=%5Cinfty
couldnt find counter example ,i thought f(x)=x but there is not g(x)
to desprove it
b) http://latex.codecogs.com/png.latex?...x%5E%7B2%7D%29 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
http://latex.codecogs.com/png.latex?...29%7D%7Bb-a%7D 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 http://latex.codecogs.com/png.latex?...f%28x%29dx%3E1
then there is http://latex.codecogs.com/png.latex?c%5Cin%28a,b%29

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

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

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.
• Aug 28th 2011, 01:30 PM
transgalactic
Re: prove or desprove claims
yes thanks :)
1 down 3 to go
• Sep 2nd 2011, 01:44 PM
transgalactic
Re: prove or desprove claims
regarding b) i was told to turn $\displaystyle 2xArctanx>ln(1+x^{2})$ into $\displaystyle 2xArctanx-ln(1+x^{2})>0$
$\displaystyle f(x)=2xArctanx-ln(1+x^{2})$
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
$\displaystyle f'(x)=2Arctanx+2\frac{1}{1+x^2}- \frac{2x}{1+x^{2}}$
how to conclude that f' is positive for x>0