Final answer:
To find the local maximum and minimum values and saddle points of the function f(x, y) = x² + xy + y² + y, we can use calculus. First, find the critical points by taking the partial derivatives of the function with respect to x and y, and setting them equal to zero. Then, evaluate the Hessian matrix and determine the type of critical point. In this case, the function has a local minimum at (-1, 2).
Step-by-step explanation:
To find the local maximum and minimum values and saddle points of the function f(x, y) = x² + xy + y² + y, we can use calculus. First, we find the critical points by taking the partial derivatives of the function with respect to x and y, and setting them equal to zero. The partial derivatives are:
∂f/∂x = 2x + y
∂f/∂y = x + 2y + 1
Setting them equal to zero, we get the equations:
2x + y = 0
x + 2y + 1 = 0
Solving these equations simultaneously, we find the critical point (-1, 2). Next, we find the Hessian matrix of second partial derivatives:
H = [[2, 1], [1, 2]]
Evaluating the determinant of this matrix, we get:
det(H) = (2)(2) - (1)(1) = 3
Since the determinant is positive, we have a local minimum at the critical point (-1, 2). Therefore, the function f(x, y) = x² + xy + y² + y has a local minimum at (-1, 2).