Question

Let $f(x)=\left|(x-1)\left(x^2-2x-3\right)\right|+x-3,x\in R$. 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

Answer: 3

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