Let $f_{1}:(0, \infty) \rightarrow \mathbb{R}$ and $f_{2}:(0, \infty) \rightarrow \mathbb{R}$ be defined by $$ f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x0 $$ and $ f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x0 $ where, for any positive integer $n$ and real numbers $a_{1},…

Question

Let $f_{1}:(0, \infty) \rightarrow \mathbb{R}$ and $f_{2}:(0, \infty) \rightarrow \mathbb{R}$ be defined by $$ f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0 $$ and $ f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0 $ where, for any positive integer $n$ and real numbers $a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}$ 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 2 m 1 + 3 n 1 + m 1 n 1 is _____.

Aicharya · Accepted Answer

Given f 1 x = &#8747; 0 x t &#8722; 1 t &#8722; 2 2 ...... t &#8722; 2 t 21 d t &#8658; &#160; &#160; &#160; f ' 1 x = x &#8722; 1 x &#8722; 2 2 ..... x &#8722; 21 21 Checking critical points At all odd integers from 1 to 21 f x will have an extrema with 1 , 5 , 9 , 13 , 17 , 21 being points of minima & 3 , 7 , 11 , 15 , 19 being points of maxima. So m 1 = 6 & n 1 = 5 Hence 2 m 1 + 3 n 1 + m 1 n 1 = 57

Let $f_{1}:(0, \infty) \rightarrow \mathbb{R}$ and $f_{2}:(0, \infty) \rightarrow \mathbb{R}$ be defined by…

Solution