Question

Let $f_1:(0,\infty )\to R$ and $f_2:(0,\infty )\to R$ be defined by $f_1(x)=\underset{0}{\overset{x}{\int }}\prod _{j=1}^{21}(t-j{)}^{j}dt,x>0$ and $f_2(x)=98(x-1{)}^{50}-600(x-1{)}^{49}+2450,x>0$, where, for any positive integer $n$ and real numbers $a_1,a_2$, $\dots ..,a_n,\prod _{i=1}^n{a}_i$ denotes the product of $a_1,a_2,\dots .,a_n.$ 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 $2m_1+3n_1+m_1{n}_1$ is _____

Answer 1

Answer: 57

$f_1(x)=\underset{0}{\overset{x}{\int }}\prod _{j=1}^{21}(t-j{)}^{j}dt$
$\Rightarrow f_1^{\prime }(x)=\prod _{j=1}^{21}(x-j{)}^j$
Therefore $m_1=6,n_1=5$
$2m_1+3n_1+m_1{n}_1$
$=2(6)+3(5)+30$
$=12+15+30=57$