For every pair of continuous functions $f, g$ $: \left[0, 1\right] \rightarrow R $, such that $max(f(x) : x \in \left[0, 1\right]) $$= $ $max( g(x) : x \in \left[0, 1\right] )$ , the correct statement(s) is(are)

(1)$(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $ c \in [0, 1] $

(2)$(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $ c \in [0, 1] $

(3)$(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $ c \in [0, 1] $

(4)$(f(c))^2 = (g(c))^2 $ for some $ c \in [0, 1] $

My work 

In this one possibility is that they have same maxima, assume it to be $c$ and this will satisfy option $1$ and $4$

Is there any other possibility than this ?