For every pair of continuous functions $f, g : [0,1] \rightarrow \mathbb{R}$ such that
$$\max\{f(x) : x \in [0,1]\} = \max\{g(x) : x \in [0,1]\}$$
the correct statement(s) is(are):\\
(A) $(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$\\
(B) $(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$\\
(C) $(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $c \in [0,1]$\\
(D) $(f(c))^2 = (g(c))^2$ for some $c \in [0,1]$