Problem related to real monic quadratic polynomial

Question

Let $f(x)$ be a real monic quadratic polynomial. If ${x_1},{x_2},{x_3},{x_4},{x_5}$ be the $5$ points where $g(x) = |f(|x|)|$ is non-differentiable and $\sum_{i = 1}^5 {\left| {{x_i}} \right| = 8} $ then find the value of $\frac{1}{5}\mathop {\lim }\limits_{x \to \infty } \frac{{{x^2} - f\left( x \right)}}{x}$.

My approach is as follow, real monic quadratic means that leading coefficient viz. the value of $a$ in $ax^2+bx+c=0$ is $1$. That is the equation is of the form $x^2+bx+c$, for real case $b^2-4c\ge0$. But not able to approach.

Answer 1

$g$ is non-differentiable at zeroes of $f$ and their reflections across $0$ and at $0$.

So if $x_1=m>0$ and $x_2=n>0$ then $x_3=-m$, $x_4=-n$ and $x_5=0$. Thus $$2n+2m =8 \implies b= -(m+n)=-4$$

So $$f(x) = x^2-4x+c \implies \lim_{x\to \infty}{4x-c\over x} = 4$$

So the result is $4/5$.

Answer 2

Hint: $$g(x)=|ax^2+b|x|+c|$$ is non-differentiable at five points if $b^2 > 4ac, b<0, c>0$.