Question

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) f(x,y)=2x 3 −6x+6xy 2 local maximum value(s) local minimum value(s) saddle point(s) (x,y)=(0,1),(0,−1)

Answer 1

The function
`\( f(x, y) = 2x^3 - 6x + 6xy^2 \)` has a local maximum value(s) at (0, 1) and (0, -1), local minimum value(s) at (0, 0), and saddle point(s) DNE.

Step-by-step explanation:

To find the critical points of the function, we need to calculate the partial derivatives of x and y and set them equal to zero. After finding the critical points, we use the second partial derivative test to determine whether each point is a local maximum, local minimum, or a saddle point.

The partial derivatives are:

`\[ f_x = 6x^2 - 6 + 6y^2 \]`

`\[ f_y = 12xy \]`

Setting
`\( f_x \)` an
`\( f_y \)` equal to zero, we find that (0, 0) is a critical point.

Using the second partial derivative test, we evaluate the discriminant
`\( D = f_(xx)f_(yy) - (f_(xy))^2 \)` at each critical point. For (0, 0), D > 0, and
`\( f_(xx) > 0 \)`, indicating a local minimum.

For the points (0, 1) and (0, -1), D < 0, suggesting saddle points.

Therefore, the function has a local minimum value(s) at (0, 0) and saddle point(s) at (0, 1) and (0, -1).