Final answer:
To find the local maxima, local minima, and saddle points of the given function, we can use the critical points and the second derivative test. After finding the critical points, we classify them using the second derivative test. We find that there is one local minimum and no local maxima or saddle points.
Step-by-step explanation:
To find the local maxima, local minima, and saddle points of the function F(x, y) = -8x² - 8xy - 9y² + 96x - 8y + 4, we need to find the critical points by taking the partial derivatives with respect to x and y, setting them equal to zero, and solving the resulting system of equations.
After finding the critical points, we can classify each point by evaluating the second-order partial derivatives using the second derivative test. If the second derivative test is inconclusive, further investigation is needed to determine if the point is a saddle point or a local extremum.
By following this process, we find that there are no local maxima, but there is one local minimum at the point (-2, 4). There are also no saddle points. So the answer to the question is:
A. A local maximum occurs at (), The local maximum value(s) is/are
B. There are no local maxima.