Which of the following is false regarding $f(x) = xe^x - 2$:
a) the function $f(x)$ is well defined and continuous for all $x$ in the interval $(0,2)$
b) the function $f(x)$ has no discontinuity and no singularities
c) the function $f'(x)$ is well defined and continuous for all $x$ in the interval $(0,2)$
d) the function $f'(x)$ has no discontinuity and no singularities
e) all of the above
When I draw the graph for both $f(x)$ and $f'(x)$ they both seem well defined and continuous on $(0,2)$ so I guess it's not a and c. However both seems to shoot up into infinity between 3 and 4. Does this mean both of them have singularities? Which in turn mean there is no correct option? I'm a bit confused.
It seems the option "None of the above" should have appeared instead of "all of the above". Both f(x) and f'(x) are continuous and well defined on (0,2). Neither of them contain any discontinuities or singularities on $\Bbb R$ so none of the options are false.