Let \( f(x) = \frac{e^x - e^{-x}}{2}, \, x \in \mathbb{R} \). Let \( f^{(k)}(a) \) denote the \( k^{th} \) derivative of \( f \) e…

Consider two functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to (1, \infty) \). Both functions are differentiab…

Consider the function \( f(x) = \frac{x^3}{3} + \frac{7}{2}x^2 + 10x + \frac{133}{2} \), \( x \in [-8, 0] \). Which of the followi…

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice-differentiable function, and suppose its second derivative satisfies \( f''(x) …

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq (x - y)^2 \) for all \( x, y \in \mathbb{R} \). Then \[…

Let \(f(x) = x^3 - 3x^2 + 2\) be a function defined on \((-1, 3]\). Which of the following statements is/are correct?