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 $\displaystyle \intop_{a}^{b}f(x)dx=1$