Question

Let $f\left(x\right)=|x^2-4x+3|$ be a function defined on $x\in [0,4]$ and $\alpha ,\beta ,\gamma$ are the abscissas of the critical points of $f\left(x\right).$ If $m$ and $M$ are the local and absolute maximum values of $f\left(x\right)$ respectively, then the value of ${\alpha }^2+{\beta }^2+{\gamma }^2+m^2+M^2$ is

Answer 1

$f\left(x\right)=|x^2-4x+3|,x\in [0,4]$

$\to$ From the graph, critical points are $(1,0),(2,1),(3,0)$
$\alpha =1,\beta =2,\gamma =3$

$\to$ Local maximum occurs at $x=2$
$\therefore$ Local maximum value, $m=f\left(2\right)=1$

$\to$ Absolute maximum value, $M=f\left(0\right)=f\left(4\right)=3$

$\therefore {\alpha }^2+{\beta }^2+{\gamma }^2+m^2+M^2=24$