Final answer:
To find the local maximum and minimum values and saddle points of the function f(x, y), calculate its partial derivatives, set them equal to zero, and evaluate the second-order partial derivatives at the critical point. To find the absolute maximum and minimum values of f(x, y) on the triangular region R, evaluate the function at the vertices of the triangle and compare the values.
Step-by-step explanation:
To find the local maximum and minimum values and saddle points of the function f(x, y), we need to calculate its partial derivatives and set them equal to zero. Let's start by finding the first-order partial derivatives:
∂f/∂x = 2x - 2
∂f/∂y = -2y + 4
Setting these derivatives equal to zero, we solve the resulting system of equations:
2x - 2 = 0 => x = 1
-2y + 4 = 0 => y = 2
These values give us a critical point (local minimum) at (1, 2).
To determine if this point is a local maximum or minimum, we need to find the second-order partial derivatives and evaluate them at the critical point. The second-order partial derivatives are:
∂²f/∂x² = 2
∂²f/∂y² = -2
∂²f/∂x∂y = 0
Using these second-order partial derivatives, we can calculate the discriminant:
D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)² = (2)(-2) - (0)² = -4
Since the discriminant is negative, we have a saddle point at (1, 2).
(b) To find the absolute maximum and minimum values of f(x, y) on the triangular region R, we need to evaluate the function at the vertices of the triangle and compare the values. The vertices of the triangle are (0, 0), (0, 2), and (2, 2).
Evaluating f(x, y) at these vertices, we get:
f(0, 0) = 0² - 0² - 2(0) + 4(0) = 0
f(0, 2) = 0² - 2² - 2(0) + 4(2) = 4
f(2, 2) = 2² - 2² - 2(2) + 4(2) = 8
Therefore, the absolute maximum value of f(x, y) on the region R is 8, which occurs at the vertex (2, 2), and the absolute minimum value is 0, which occurs at the vertex (0, 0).