Final answer:
Lagrange multipliers are used to find maxima and minima of f(x, y) = y² - x² given the constraint 1/4 x² + y² = 1 by solving a system of equations that equates the gradients of the function and the constraint.
Step-by-step explanation:
To find the maximum and minimum values of the function f(x, y) = y² - x² subject to the constraint 1/4 x² + y² = 1, we can use the method of Lagrange multipliers. This method involves introducing a new variable, λ (the Lagrange multiplier), and solving the system of equations derived from the gradient of our function and the gradient of our constraint equation being proportional, i.e., ∇f(x, y) = λ ∇g(x, y), where g(x, y) represents the constraint equation.
To apply Lagrange multipliers here, we start by setting up the following system of equations:
∇f(x, y) = ∇(λ g(x, y))
2y = 2λy
-2x = 1/2λx
1/4 x² + y² = 1
We then solve this system to find possible points (x, y) that satisfy both the function and the constraint, which will give us the maximum and minimum values of f under the constraint.