Question

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by $f_{1}(x)=\int\limits_{0}^{x} \displaystyle\prod_{j=1}^{21}(t-j)^{j} d t, x>0$ and $f _{2}( x )=98( x -1)^{50}-600( x -1)^{49}+2450, x >0$, where, for any positive integer $n$ and real numbers $a _{1}, a _{2}$, $\ldots . ., a_{n}, \displaystyle\prod_{i=1}^{n} a_{i}$ denotes the product of $a_{1}, a_{2}, \ldots ., a_{n} .$ Let $m_{i}$ and $n_{i}$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _{ i }, i =1,2$, in the interval $(0$, $\infty$ ).
The value of $6 m_{2}+4 n_{2}+8 m_{2} n_{2}$ is _____

Answer 1

$f _{2}^{\prime}( x )=(98)(50)( x -1)^{49}-(600)(49)( x -1)^{48}$
$=(49)(100)( x -1)^{48}( x -1-6)$
$m _{2}=1, n _{2}=0$
$6 m _{2}+4 n _{2}+8 m _{2} n _{2}$
$6(1)+4(0)+8(0)=6$