Final answer:
To find the extremes of the function f(x, y) under a constraint, we use the method of Lagrange multipliers to set up a system of equations derived from the gradients of the function and the constraint. The solutions to this system of equations yield the critical points that are tested to determine the maximum and minimum values.
Step-by-step explanation:
To find the maximum and minimum values of the function f(x, y)=x^{2}+7y subject to the constraint x^{2}-y^{2}=7 using Lagrange multipliers, we set up the Lagrangian function L(x, y, λ) = f(x, y) - λ(g(x, y) - c), where g(x, y) = x^{2} - y^{2} is the constraint and c=7. Then, we look for where the gradient of f is parallel to the gradient of g, by solving the equations that result from setting the gradient of L to zero. In general, you will get a system of equations that you need to solve for x, y, and λ.
Calculating the partial derivatives of L with respect to x, y, and λ and setting them equal to zero, provides the necessary critical points. These are then tested in the original function f(x, y) to determine which of them are maximum or minimum, based on the given constraint g(x, y) = 7.