Final answer:
With only partial derivatives and a point provided, but no explicit function f(x, y), it is impossible to determine if the point (1, -1) is a relative maximum, saddle point, relative minimum, or none of these without additional information.
Step-by-step explanation:
The question given by the student involves multivariable calculus, specifically about characterizing the nature of a point on a function of two variables. We are given the partial derivatives fₓ (x,y) = 2(x−1) cos(πy) and fₗ(x,y) = π(2x−x²) sin(πy). Without an explicit function f(x, y), we can't directly apply the second derivative test. However, the mention of f(1,−1) suggests that we need to evaluate the partial derivatives at the point (1, −1), and analyze the values to infer the nature of this point on the graph of f(x, y). Due to the absence of the actual function f and lacking information, there is no sufficient data to conclude whether (1, -1) is a relative maximum, saddle point, relative minimum, or none of these options. More information is required for a definitive answer.