Final answer:
To find the absolute extrema for f(x,y) on the region R, we analyze the critical points and the boundary of the region. The critical point is (0, 1) and the extrema on the boundary occur where f(x,y) = 5 - 2y is minimized or maximized.
Step-by-step explanation:
In order to find the absolute extrema for f(x,y) on the region R, we need to analyze the critical points and the boundary of the region.
Step 1: Find the partial derivatives of f(x,y) with respect to x and y.
∂f/∂x = 2x and ∂f/∂y = 2y - 2
Step 2: Find the critical points by setting the partial derivatives equal to zero.
Setting ∂f/∂x = 0 gives us 2x = 0, which implies x = 0.
Setting ∂f/∂y = 0 gives us 2y - 2 = 0, which implies y = 1.
Therefore, the critical point is (0, 1).
Step 3: Analyze the boundary of the region R.
Since R is defined as x² + y² < 4, we need to determine the points on the boundary of R where f(x,y) could potentially have extreme values.
On the boundary of R, we have x² + y² = 4.
Substituting this into f(x,y), we get f(x,y) = 4 - 2y + 1 = 5 - 2y.
So the absolute extrema for f on R occur at the critical point (0, 1) and at the points on the boundary where f(x,y) = 5 - 2y is minimized or maximized.