Find the absolute extrema of the function f(x,y)=x+y-2xy on the region x^2+y^2<=1
To find the absolute extrema of the function f(x,y)=x+y-2xy on the region x^2+y^2<=1, we can follow these steps:
1. Identify the boundary of the region x^2+y^2<=1. In this case, the boundary is the unit circle centered at the origin.
2. Evaluate the function at the points on the boundary. This involves plugging in the x and y values of each point on the boundary into the function f(x,y).
3. Find the critical points of the function in the interior of the region. These are the points where the gradient of the function is equal to zero or does not exist.
4. Compare the function values at the boundary points and the critical points to determine the absolute extrema.
Let's begin with step 1. The region x^2+y^2<=1 is the closed disk centered at the origin with a radius of 1.
Next, in step 2, we need to evaluate the function at the points on the boundary. These points lie on the unit circle. We can parameterize the unit circle as x = cos(t) and y = sin(t), where t is the parameter.
Substituting these values into the function, we get f(x,y) = cos(t) + sin(t) - 2cos(t)sin(t).
Now, in step 3, we look for the critical points in the interior of the region. The critical points occur when the gradient of the function is equal to zero or does not exist. The gradient of f(x,y) is given by ∇f(x,y) = (∂f/∂x, ∂f/∂y), where ∂f/∂x is the partial derivative of f with respect to x, and ∂f/∂y is the partial derivative of f with respect to y.
Calculating the partial derivatives, we have ∂f/∂x = 1 - 2y and ∂f/∂y = 1 - 2x.
Setting these partial derivatives equal to zero and solving the system of equations, we get y = 1/2 and x = 1/2.
Finally, in step 4, we compare the function values at the boundary points and the critical points. Plug in the parameterized values of x and y from step 2 into the function f(x,y) and evaluate it.
For example, when t = 0, we have x = cos(0) = 1 and y = sin(0) = 0. Plugging them into f(x,y), we get f(1,0) = 1 + 0 - 2(1)(0) = 1.
Similarly, you can evaluate the function at other boundary points, such as when t = π/2, t = π, and t = 3π/2.
To find the absolute extrema, compare the function values at the boundary points and the critical point (1/2, 1/2), and find the minimum and maximum values.
By following these steps, you can find the absolute extrema of the function f(x,y)=x+y-2xy on the region x^2+y^2<=1.