Question

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

Hint:

To find the maximum and minimum of a function on a constrained set, parametrize the constraint and then maximize or minimize the function.

Answer 1

Correct Answer: 10.1

We are given that \( x^2 + y^2 = 1 \), so we use this constraint to find the maximum and minimum values of \( f(x, y) \). By substituting \( x = \cos \theta \) and \( y = \sin \theta \), we can rewrite the function \( f(x, y) \) as: \[ f(x, y) = 8 \cos^2 \theta - 2 \sin \theta. \] Maximizing and minimizing this expression gives \( M - m \approx 10.13 \). Thus, the value is \( 10.13 \).