List of top Statistics Questions on Optimization

Let \( f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x, y) = 8x^2 - 2y, \text{ where } \mathbb{R} \text{ denotes the set of all real numbers.} \] If \( M \) and \( m \) denote the maximum and minimum values of \( f \), respectively, on the set \[ \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}, \] then M - m = _________ (round off to 2 decimal places

Let \( f : [0, \frac{\pi}{2}] \to \mathbb{R} \) be defined as \[ f(x) = ax + \beta \sin x, \] where \( a, \beta \in \mathbb{R} \). Let \( f \) have a local minimum at \( x = \frac{\pi}{4} \) with \[ f' \left( \frac{\pi}{4} \right) = -\frac{4}{\sqrt{2}}. \] Then \( 8\sqrt{2a} + 4 \beta \) equals