Question

[ f(x, y)=x³-48 x y+64 y³] local maximum value(s) local minimum value(s) saddle point(s)

Answer 1

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.