Find the absolute maximum and minimum values of the following function on the specified region R.
F(x, y) = 9xy on the semicircular disk.
R = {(x, y): −2 ≤ x ≤ 2, 0 ≤ y ≤ (4 − x²):}

Step-by-step explanation:

Since it is said that the region R is a semicircular disc, we asume that the boundaries of the region are given by . First, we must solve the optimization problem without any restrictions, and see if the points we get lay inside the region of interest. To do so, consider the given function F(x,y). We want to find the point for which it's gradient is equal to zero, that is

This implies that (x,y) = (0,0). This point lays inside the region R. We will use the Hessian criteria to check if its a minimum o r a maximum. To do so, we calculate the matrix of second derivates

so we get the matrix

Note that the first determinant is 0, and the second determinant is -9. THis tell us that the point  is a saddle point, hence not a minimum nor maximum.  

Since the function is continous and the region R is closed and bound (hence compact) the maximum and minimum must be attained on the boundaries of R. REcall that when and y=0 we have that F(x,0) = 0. So, we want to pay attention to the critical values over the circle, restricting that the values of y must be positive. To do so, consider the following function

which consists of the original function and a function that describes the restriction (the circle x^2+y^2=4), we want that the gradient of H is 0.

Then,

From the first and second equation we get that

which implies that . If we replace this in the restriction, we have that which gives us that . Since we only care for the positive values of y, and that , we have the following critical points . Note that for the first point, the value of the function is

as for the second point the value of the function is

.

Then, the point is a maximum and the point is a minimum.

squares, rhombus, rectangles