Question

Let $f(x)=\left|(x-1)\left(x^{2}-2 x-3\right)\right|+x-3, x \in R$. If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $(0,4)$, then $m+M$ is equal to_____

Answer 1

$f(x)=\begin{cases}\left(x^{2}-1\right)(x-3)+(x-3), x \in(0,1] \cup[3,4) \\-\left(x^{2}-1\right)(x-3)+(x-3), x \in[1,3]\end{cases}$
$\Rightarrow f^{\prime}(x)=\begin{cases}3 x^{2}-6 x, x \in(0,1) \cup(3,4) \\-3 x^{2}+6 x+2, x \in(1,3)\end{cases}$
$f ( x )$ is non-derivable at $x =1$ and $x =3$
also $f^{\prime}(x)=0$ at $x=1+\sqrt{\frac{5}{3}}$
$\Rightarrow m+M=3$