For every pair of continuous functions $$f, g: [0, 1] \rightarrow R$$ such that
$$max \left\{f(x) : x \in [0, 1]\right\} = max \left\{{g(x) : x \in [0, 1]}\right\}$$
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 $$c \in [0, 1]$$