Question

Find the local maximum and minimum values and saddle points of the function.

a) f(x,y) = x² +xy+y² +y
b) f(x,y)=xy−x² y−xy²

Answer 1

To determine local maxima, minima, and saddle points of the given functions, one should find the partial derivatives, critical points, and use the Second Derivative Test.

Step-by-step explanation:

To find the local maximum, minimum values, and saddle points of a function of two variables, we use the method of finding critical points and analyzing the second partial derivatives, often using the Second Derivative Test. This involves finding the first partial derivatives ∂f/∂x and ∂f/∂y, setting them equal to zero to locate critical points, and then using the second partial derivatives to analyze these points.

For the function f(x,y) = x² + xy + y² + y, we would first find the partial derivatives with respect to x and y, set them to zero to find critical points, and use the second partial derivatives to apply the Second Derivative Test.

Similarly, for the function f(x,y) = xy - x²y - xy², the process would be the same: find the partial derivatives, determine the critical points, and apply the Second Derivative Test to classify them as local maxima, minima, or saddle points.