Let $f$ be the function defined for $x > 0$, with $f(e) = 2$ and $f'$, the first derivative of $f$, given by $f'(x) = x^2 \ln x$.
(a) Write an equation for the line tangent to the graph of $f$ at the point $(e, 2)$.
(b) Is the graph of $f$ concave up or concave down on the interval $1 < x < 3$? Give a reason for your answer.
(c) Use antidifferentiation to find $f(x)$.