Question

Find all the local maxima, local minima, and saddle points of the function f(x,y) = 2x2 - 4xy + 3y2 - 8x + 3y + 5 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice OA. A local maximum occurs at (Type an ordered pair Use a comma to separate answers as needed) The local maximum value(s) is/are (Type an exact answer. Use a comma to separate answers as needed) OB. There are no local maxima Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice O A. Alocal minimum occurs at (Type an ordered pair Use a comma to separate answers as needed) The local minimum value(s) is/are (Type an exact answer Use a comma to separate answers as needed) OB. There are no local minima Select the correct choice below and, if necessary fill in the answer box to complete your choice OA. A saddle point occurs at (Type an ordered pair Use a comma to separate answers as needed.)

Answer 1

To find the local maxima, local minima, and saddle points of the function f(x, y) = 2x^2 - 4xy + 3y^2 - 8x + 3y + 5, we can calculate the critical points and use the second derivative test.

First, let's find the partial derivatives:

∂f/∂x = 4x - 4y - 8

∂f/∂y = -4x + 6y + 3

To find the critical points, we set both partial derivatives equal to zero:

4x - 4y - 8 = 0

-4x + 6y + 3 = 0

Solving these equations simultaneously, we get:

x = 1

y = -1

Now, let's calculate the second partial derivatives:

∂^2f/∂x^2 = 4

∂^2f/∂y^2 = 6

∂^2f/∂x∂y = -4

Using these second partial derivatives, we can calculate the discriminant to determine the nature of the critical point:

D = (∂^2f/∂x^2) * (∂^2f/∂y^2) - (∂^2f/∂x∂y)^2

Plugging in the values, we have:

D = (4)(6) - (-4)^2

D = 24 - 16

D = 8

Since D is positive and (∂^2f/∂x^2) is positive, the critical point (1, -1) corresponds to a local minimum.

Therefore, the correct choices are:

OA. A local minimum occurs at (1, -1). The local minimum value is unknown without further calculation.

OB. There are no local maxima.

OA. A saddle point occurs at (1, -1).