Critical Points & their Nature of a Multivariable Function

[wowmaths]

Homework Statement

f(x,y) = xy(9x^2 + 3y^2 -16)

Find the critical points of the function and their nature (local maximum, local minimum or saddle)

Homework Equations

The Attempt at a Solution

I have partially differentiated the equation into:

fx = 27yx^2* + 3y^3 -16y
fy = 9x^3 + 9xy^2 - 16x

How do I go on from there though?

 

[STEMucator]

Hey there, welcome to the forum.

So, recall that a critical point of a function f(x) occurs when the derivative f'(x) = 0.

Now translating that to the multivariate case is not very different. Set f_x = 0 and f_y = 0 and then solve for your critical points.

 

[Ray Vickson]

Homework Statement

f(x,y) = xy(9x^2 + 3y^2 -16)

Find the critical points of the function and their nature (local maximum, local minimum or saddle)

Homework Equations

The Attempt at a Solution

I have partially differentiated the equation into:

fx = 27yx^2* + 3y^3 -16y
fy = 9x^3 + 9xy^2 - 16x

How do I go on from there though?

You solve the equations
27*y*x^2 + 3*y^3 - 16*y = 0
9*x^3 + 9*x*y^2 - 16*x = 0.
The first one says either y = 0 or 27*x^2 + 3*y^2 - 16 = 0, and this last one is the equation of an ellipse in (x,y) space. The second one says either x = 0 or 9*x^2 + 9*y^2 -16 = 0. What kind of geometric figure does that describe in (x,y)-space?

RGV

 

[wowmaths]

Hey there, welcome to the forum.

So, recall that a critical point of a function f(x) occurs when the derivative f'(x) = 0.

Now translating that to the multivariate case is not very different. Set f_x = 0 and f_y = 0 and then solve for your critical points.

You solve the equations
27*y*x^2 + 3*y^3 - 16*y = 0
9*x^3 + 9*x*y^2 - 16*x = 0.
The first one says either y = 0 or 27*x^2 + 3*y^2 - 16 = 0, and this last one is the equation of an ellipse in (x,y) space. The second one says either x = 0 or 9*x^2 + 9*y^2 -16 = 0. What kind of geometric figure does that describe in (x,y)-space?

RGV

Thanks a lot for the welcome and the help. Greatly appreciated.
Just a question:
For both the equations:
fx = 27*y*x^2 + 3*y^3 -16*y = 0
and
fy = 9*x^3 + 9*x*y^2 - 16*x = 0

Am I supposed to solve both equations simultaneously to find the stationary points?

I tried using a calculator to solve both equations simultaneously and I've gotten 9 stationary points.

Am I on the right track?

 

[STEMucator]

Ill give you a hint, use elimination to solve both. Solve for y in the first equation to get y=0 and y = ± something else.

 

[Ray Vickson]

Thanks a lot for the welcome and the help. Greatly appreciated.
Just a question:
For both the equations:
fx = 27*y*x^2 + 3*y^3 -16*y = 0
and
fy = 9*x^3 + 9*x*y^2 - 16*x = 0

Am I supposed to solve both equations simultaneously to find the stationary points?

I tried using a calculator to solve both equations simultaneously and I've gotten 9 stationary points.

Am I on the right track?

Answer to both questions is yes. The geometric representation I suggested before shows why there are 9 solution points.

RGV