Question

Find the absolute maximum and minimum values of the functionf(x, y) = x ^2 − 2xy + 2y on the rectangle D = (x, y)

a) (0,0)
b) (1,1)
c) (2,0)
d) (3,2

Answer 1

To find the maximum and minimum of the function f(x, y) on the given rectangle, calculate the partial derivatives, find critical points, and evaluate the function at the corners of the rectangle.

Step-by-step explanation:

To find the absolute maximum and minimum values of the function f(x, y) = x^2 - 2xy + 2y on the rectangle D = 0 ≤ x ≤ 3, 0 ≤ y ≤ 2, we need to consider both the interior and the boundary of the rectangle.

First, we calculate the partial derivatives of f with respect to x and y:

• fx(x, y) = 2x - 2y
• fy(x, y) = 2 - 2x

Setting these partial derivatives equal to zero gives us the critical points of the function in the interior of the rectangle:

• 2x - 2y = 0 → x = y
• 2 - 2x = 0 → x = 1

The only critical point inside the domain is (1,1) because when x = y, x has to equal 1 to satisfy both equations.

Next, we evaluate f at the corners of the rectangle D, which are (0,0), (3,0), (0,2), and (3,2), and compare these values to f(1,1). After calculation, we will identify the largest and smallest values.