Which of the following is correct option? 
Let $f_1,f_2$ be two real  valued functions defined on the real
  line.Define two functions $g,h$ by $g(x)=max${$f_1(x),f_2(x)$} and
  $h(x)=min${$f_1(x),f_2(x)$}.Then
  $g(x^2)+h(x)^2+3g(x)h(x)=f_1(x)^2+f_2(x)^2+3f_1(x)f_2(x)$ holds for
  all $x\in \mathbb R$
$1.$Always
$2.$ only of $f_1(x)=f_2(x)$ for all $x\in \mathbb R$
$3.$only $f_1$ and $f_2$ are both positive functions or both negative
  functions.
$4.$ only if atleast one of the functions $f_1$ and $f_2$ is
  identically zero

Solution:
if we choose $f_1(x)=1,f_2(x)=2$,then  $g(x^2)+h(x)^2+3g(x)h(x)=2+1^2+3(2)(1)=9$ 
and 
$f_1(x)^2+f_2(x)^2+3f_1(x)f_2(x)=1^2+2^2+3(2)(1)=11$,
So,
$g(x^2)+h(x)^2+3g(x)h(x)\neq f_1(x)^2+f_2(x)^2+3f_1(x)f_2(x)$
Hence,option $1,3,4$ get discarded so,the only remaining option i.e option 2.
must be true.
But, it is given that option $1$ is true.
Please point out my mistake 

Hint, assuming my comment above is true: Note that $f_1(x)+f_2(x)=g(x)+h(x)$ and $f_1(x)f_2(x)=g(x)h(x)$.
