Which of the following is correct option?

Question

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

Answer 1

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