3 questions from 2021
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = \dfrac{x^{2} - 3x - 6}{x^{2} + 2x + 4} \). Then which of the following statements is (are) TRUE?
Let \( f_1 : (0,\infty) \to \mathbb{R} \) and \( f_2 : (0,\infty) \to \mathbb{R} \) be defined by
\( f_1(x) = \displaystyle\int\limits_{0}^{x} \prod_{j=1}^{21} (t - j)^{j} \, dt, \quad x > 0 \) 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 maximum of function \( f_i, \, i=1,2, \) in the interval \( (0,\infty) \).
The value of \( \displaystyle 6m_2 + 4n_2 + 8m_2n_2 \) is ________.
Let \( f_1 : (0,\infty) \to \mathbb{R} \) and \( f_2 : (0,\infty) \to \mathbb{R} \) be defined by
\( f_1(x) = \displaystyle\int\limits_{0}^{x} \prod_{j=1}^{21} (t - j)^{j} \, dt, \quad x > 0 \) 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 maximum of function \( f_i, \, i=1,2, \) in the interval \( (0,\infty) \).
The value of \( 2m_1 + 3n_1 + m_1 n_1 \) is ________.
