To solve for the values of x and y that satisfy the equations fxx(x, y) = 0 and fyx(x, y) = 0 simultaneously, we begin by understanding the function provided:
f(x, y) = x2 + 4xy + y2 – 34x – 32y + 38
Next, we need to compute the second partial derivatives:
- First, we find the first partial derivatives:
- fx(x, y) = 2x + 4y – 34
- fy(x, y) = 4x + 2y – 32
Then, we compute the second partial derivatives:
- fxx(x, y) = 2
- fyx(x, y) = 4
For fxx(x, y) = 0, we see that this is not satisfied since it is constantly 2. Therefore, we cannot find any values of x and y that would make this true.
On the other hand, for fyx(x, y) = 0, it is constantly 4, which similarly cannot be made to equal 0.
Given that both second partial derivatives are constants and never equal to zero, we conclude that there are no values of x and y that satisfy the equations simultaneously.