Final answer:
The question involves finding the maximum and minimum values of the function f(x,y)=x-2y given the constraint x² + 2y² = 1, using the method of Lagrange multipliers.
Step-by-step explanation:
The student is asking to find the extreme values of the function f(x,y)=x-2y subject to the constraint x² + 2y² = 1. This is a problem of finding the maximum and minimum of a function of two variables with a given constraint, which is typically approached using the method of Lagrange multipliers. This method involves introducing an auxiliary function called the Lagrangian, L(x,y,λ) = f(x,y) - λ(g(x,y)-c), where g(x,y) is the constraint equation and c is the constraint value. The extreme values are found by taking the partial derivatives of L with respect to x, y, and λ and setting them equal to zero. These equations are then solved simultaneously to find the critical points. After finding the critical points, the original function f(x,y) is evaluated at these points to determine the extreme values.