A real coefficient polynomial $f(x)$ has degree greater than 5, and its leading coefficient is positive. Moreover, $f(x)$ has local minima at $x = 1, 2, 4$ and local maxima at $x = 3, 5$. Based on the above, select the correct options.
(1) $f(1) < f(3)$
(2) There exist real numbers $a, b$ satisfying $1 < a < b < 2$ such that $f'(a) > 0$ and $f'(b) < 0$
(3) $f''(3) > 0$
(4) There exists a real number $c > 5$ such that $f'(c) > 0$
(5) The degree of $f(x)$ is greater than 7