MHB Critical Points & Extrema of f (x, y)

[harpazo]

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails.

f (x, y) = x^3 + y^3

Solution:

f_x = 3x^2

f_y = 3y^2

f_xx = 6x

f_th = 6y

f_2xy = 0

I set f_x and f_y = 0 and found the critical points to be
(0, 0).

Is this right so far?

d = (6x)(6y) - [0]^2

d = 36xy

d = 36 (0)(0) - 0

d = 0

Since d = 0, the test is inconclusive.

I evaluated f (x, y) at (0, 0) and found the point in space to be (0, 0, 0).

The textbook goes on to say that the test for relative extrema fails and that we have a saddle point here.

1. Why does the test fails here?

2. I am not too sure how to show that we have a saddle point in this case.

Can someone explain this question in simple terms?

 

[Prove It]

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails.

f (x, y) = x^3 + y^3

Solution:

f_x = 3x^2

f_y = 3y^2

f_xx = 6x

f_th = 6y

f_2xy = 0

I set f_x and f_y = 0 and found the critical points to be
(0, 0).

Is this right so far?

d = (6x)(6y) - [0]^2

d = 36xy

d = 36 (0)(0) - 0

d = 0

Since d = 0, the test is inconclusive.

I evaluated f (x, y) at (0, 0) and found the point in space to be (0, 0, 0).

The textbook goes on to say that the test for relative extrema fails and that we have a saddle point here.

1. Why does the test fails here?

2. I am not too sure how to show that we have a saddle point in this case.

Can someone explain this question in simple terms?

In 3D a saddle point is the equivalent of a stationary point of inflection in 2D. As this function is the sum of two functions which have SPOIs at (0,0) it makes sense that in 3D you would have a saddle point.

As for proving it, you need to show that movement in some directions from that point results in a decrease while movement in other directions from that point results in an increase. That would be enough to show that it is not a maximum or a minimum, and thus is a saddle point.