certain properties of a function

it was an example in a lecture:

f is continues in R. there is certain "a" for which f(a)=0.

and there exists b which a<b and f(b)<0 and also f'(a)>0

is there another root to f(x)

prove or show a dissproving example

__i was given a dissproving example like this:__

we take f(x)

f(x)=$\displaystyle x^2sin(\frac{1}{x^2})$ when x differ 0

f(x)=0 for x=0

he says some thing about that this is an example to a function which is differentiable

but its derivetive... (i coulnd ubderstand its special property )

??

then he says

we define g(x)=f(x)+x

and this is our desproving example

how he got to this g(x)

??

why its desproving

??