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

$f, g : [0,1] \to \mathbb{R} \text{ continuous}$

$\max_{[0,1]} f = \max_{[0,1]} g$

$\text{To find: which statement(s) is(are) always true}$

$\text{Let } \max f = \max g = M \text{ (attained at } x_1, x_2)$

$h=f-g: \ h(x_1)\ge 0, \ h(x_2)\le 0 \Rightarrow \exists\, c \in [0,1]:\ f(c)=g(c)$

$f(c)=g(c)=t \Rightarrow \Phi(f(c)) = \Phi(g(c)) \text{ for the same } \Phi$

$(1)\ t^2+3t = t^2+3t \ \checkmark, \qquad (4)\ t^2 = t^2 \ \checkmark$

$(2)\ f=x,\ g=1:\quad (f(c))^2+f(c) = c^2+c, \quad (g(c))^2+3g(c) = 4$

$c^2+c-4=0 \Rightarrow c = \dfrac{-1+\sqrt{17}}{2} \approx 1.56 \notin [0,1] \Rightarrow \text{false}$

$(3)\ f=1,\ g=x:\quad (f(c))^2+3f(c) = 4, \quad (g(c))^2+g(c) = c^2+c \le 2$

$4 = c^2+c \text{ has no solution in } [0,1] \Rightarrow \text{false}$

$\boxed{\text{Correct statements: (1) and (4)}}$