Question

Let $f\left(x\right)=(x-1{)}^m(2-x{)}^n;m,nϵN$ and $m,n>2$.

Answer 1

$y=(x-1{)}^m(2-x{)}^n$
$y^{\prime }=m(x-1{)}^{m-1}(2-x{)}^n-n(2-x{)}^{n-1}(x-1{)}^m$
$=(x-1{)}^{m-1}(2-x{)}^{n-1}[2m-mx-nx+n]$
$=(x-1{)}^{m-1}(2-x{)}^{n-1}[2m+n-(m+n\left)x\right]$
Considering $n$ is even we get
$y^{\prime }=m(x-1{)}^{m-1}(x-2{)}^n+n(x-2{)}^{n-1}(x-1{)}^m$
$=(x-1{)}^{m-1}(2-x{)}^{n-1}[mx-2m+nx-n]$
$=(x-1{)}^{m-1}(2-x{)}^{n-1}\left[\right(m+n)x-(2m+n\left)\right]$
Hence for both m and n odd, we get 1 and 2 as the inflection points.
For both even we get 1 and 2 as points of minima.
Hence
$A\to 1,3,5$
$B\to 1,4$
$C\to 2,3$
$D\to 2,4$