Question

Find the critical point of the function f(x,y)=28x−4y2−ln(∣x+y∣) c= Use the Second Derivative Test to determine whether it is

A. a local minimum
B. a saddle point
C. test fails
D. a local maximum

Answer 1

B. a saddle point

The critical point, determined by setting the partial derivatives to zero, results in a saddle point as indicated by the Second Derivative Test.

Step-by-step explanation:

To find the critical point, we need to find where the partial derivatives of the function f(x, y) are equal to zero. After finding the critical point, we use the Second Derivative Test to determine the nature of the critical point.

The critical point for the given function is determined by setting the partial derivatives
`(f_x)` and
`(f_y)` equal to zero and solving the system of equations. Once the critical point is identified, we can use the Second Derivative Test. If the determinant of the Hessian matrix is positive and the second partial derivative with respect to \(x\) is positive at the critical point, it indicates a local minimum. If the determinant is negative, it indicates a local maximum. However, if the determinant is zero, the test fails to provide conclusive information.

In this case, the Second Derivative Test indicates a saddle point because the determinant is negative, suggesting the presence of both concave-up and concave-down directions.