Question

Consider the function f(x,y)=(8x−x2)(4y−y2).

Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank.
fx =
fy=
fex =
fxy =
fyy =

Answer 1

To find and classify the critical points of the function f(x,y), we calculate the first-order partial derivatives fx and fy, set them to zero to find potential critical points, and then use second-order partial derivatives f_xx, fxy, and fyy along with the Hessian determinant for classification.

Step-by-step explanation:

We are asked to find and classify the critical points of the function f(x,y)=(8x-x^2)(4y-y^2). To do this, we begin by finding the first-order partial derivatives of the function, fx and fy.

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

• fx = 0
• fy = 0

Then, we solve the system of equations for x and y. After finding the potential critical points, we classify them using the second-order partial derivatives:

• fx_x (second partial derivative with respect to x)
• fxy (mixed partial derivative)
• fyy (second partial derivative with respect to y)

We use the second derivative test by computing the determinant of the Hessian matrix, D = f_xxfyy - (fxy)^2. If D > 0 and fx_x > 0, then (x, y) is a local minimum. If D > 0 and f_xx < 0, then (x, y) is a local maximum. If D < 0, then (x, y) is a saddle point.