## 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)= $x^2sin(\frac{1}{x^2})$ 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
??