Final answer:
To find the local maximum, minimum, and saddle points of f(x, y) = xy - e^(-x^2-y^2), we find the critical points by setting the partial derivatives equal to zero.
Step-by-step explanation:
The function f(x, y) = xy - e^(-x^2-y^2) has critical points where the partial derivatives are equal to zero. To find these critical points, we find the partial derivatives:
∂f/∂x = y + 2xe^(-x^2-y^2) = 0, and ∂f/∂y = x + 2ye^(-x^2-y^2) = 0.
We can solve these equations to find the critical points, and then use the second partial derivatives test to determine if they are local maxima, local minima, or saddle points.