JEE Advanced 2021 · Paper 2, Mathematics question 9

Question Stem for Question Nos. 9 and 10

Let $f_1: (0, \infty) \to \mathbb R$ and $f_2: (0, \infty) \to \mathbb R$ be defined by
$$f_1(x) = \int_0^x \prod_{j=1}^{21}(t - j)^j\,dt,\qquad x > 0$$
and
$$f_2(x) = 98(x - 1)^{50} - 600(x - 1)^{49} + 2450,\qquad x > 0,$$
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 $(0, \infty)$.