Final answer:
To find the local maximum and minimum values, as well as saddle points, of the function f(x, y) = x³ - 48xy + 64y³, we need to find the critical points and then use the Second Partial Derivative Test.
Step-by-step explanation:
To find the local maximum and minimum values, as well as saddle points, of the function f(x, y) = x³ - 48xy + 64y³, we need to find the critical points and then use the Second Partial Derivative Test.
Step 1: Find the first-order partial derivatives of f(x, y).
Step 2: Find the critical points by solving the system of equations obtained by setting the first-order partial derivatives equal to zero.
Step 3: Find the second-order partial derivatives of f(x, y).
Step 4: Evaluate the second-order partial derivatives at the critical points and use the Second Partial Derivative Test to determine the nature of each critical point.