Question

What is the maximum and minimum values of the function f(x, y) = y² - x² subject to the constraint 1/4 * x² + y² = 49?

Answer 1

The maxima and minima of the function f(x, y) = y² - x² with constraint 1/4 * x² + y² = 49 can be obtained using the method of Lagrange multipliers. This involves forming a Lagrange function, taking partial derivatives, setting them to zero to find candidate points, and evaluating those points in the original function.

Step-by-step explanation:

The maximum and minimum values of the function f(x, y) = y² - x² subject to the constraint 1/4 * x² + y² = 49 can be found using the method of Lagrange multipliers. This is a constraint optimization problem where one has to find the extreme values of the function f(x, y) with the given constraint.

First, we form the Lagrange function L(x, y, λ) = y² - x² + λ(1/4 * x² + y² - 49). Taking the partial derivatives of L with respect to x, y, and λ and setting them equal to zero gives us a system of equations:

1. ∂L/∂x = -2x + λ*x/2 = 0
2. ∂L/∂y = 2y + 2λ*y = 0
3. ∂L/∂λ = 1/4 * x² + y² - 49 = 0

By solving this system, we can find the candidate points for extreme values, which then can be evaluated by substituting back into the function f(x, y) to check for maximum or minimum values.