f(x,y)=x⁴+2y² + 16xy
The critical point(s) is/are (0,0),(4,-16), (-4,16).
(Type an ordered pair. Use a comma to separate answers as needed.)
Use the Second Derivative Test to find the local maxima. Select the correct choice and, if necessary, fill in the answer box to
complete your choice.
A. The test shows that the local maximum (maxima) is (are) at
(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)
B. The test does not reveal any local maxima and there are no critical points for which the test is inconclusive, so there are
no local maxima.
OC. The test does not reveal any local maxima, but there is at least one critical point for which the test is inconclusive.
Use the Second Derivative Test to find the local minima. Select the correct choice and, if necessary, fill in the answer box to
complete your choice.
A. The test shows that the local minimum (minima) is (are) at (4,-16), (-4,16).
(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)
OB. The test does not reveal any local minima and there are no critical points for which the test is inconclusive, so there are
no local minima.
OC. The test does not reveal any local minima, but there is at least one critical point for which the test is inconclusive.
Use the Second Derivative Test to find the saddle points. Select the correct choice and, if necessary, fill in the answer box to
complete your choice.
A. The test shows that the saddle point(s) is/are at (0,0).
(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)
OB. The test does not reveal any saddle points and there are no critical points for which the test is inconclusive, so there are
no saddle points.
OC. The test does not reveal any saddle points, but there is at least one critical point for which the test is inconclusive.
Determine the behavior of the function at any of the critical points for which the Second Derivative Test is inconclusive. Select
the correct choice and, if necessary, fill in the answer box(es) to complete your choice.
(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)
A. Among these points, there are local maximum/maxima at
local minimum/minima at and saddle point(s) at
B. Among these points, there are local maximum/maxima at
no local minima, and no saddle points.
local minimum/minima at and no saddle points.
no local minima, and saddle point(s) at
OC. Among these points, there are local maximum/maxima at
OD. Among these points, there are local maximum/maxima at
E. Among these points, there are no local maxima, local minimum/minima at
OF. Among these points, there are no local maxima, local minimum/minima at
OG. Among these points, there are no local maxima, no local minima, and saddle
OH. Among these points, there are no local maxima, no local minima, and no saddle points.
1. The Second Derivative Test is conclusive for each critical point.
and saddle point(s) at
and no saddle points.
point(s) at
